Sound, Physics and Music

1. Basic Concepts
1. Talking about Sound and Music
. Transverse and Longitudinal Waves
. Frequency, Wavelength, and Pitch
. Sound Amplitude and Musical Dynamics
. Presenting Concepts to Younger Students
1. Sound and Music Activities
2. Sound and Ears
2. The Physics and Math of Intervals and Tuning
1. Musical Intervals, Frequency, and Ratio
2. Harmonic Series

wu BW N

4. Tuning Systems

3. Standing Waves and Instruments
1. Standing Waves and Musical Instruments
2. Standing Waves and Wind Instruments

Talking about Sound and Music
For middle school and up, an introduction to some acoustics terms and how
they relate to music.

Music is the art of sound, so let's start by talking about sound. Sound is
invisible waves moving through the air around us. In the same way that
ocean waves are made of ocean water, sound waves are made of the air (or
water or whatever) they are moving through. When something vibrates, it
disturbs the air molecules around it. The disturbance moves through the air
in waves - each vibration making its own wave in the air - spreading out
from the thing that made the sound, just as water waves spread out from a
stone that's been dropped into a pond. You can see a short animation of a
noise being created here.

Surf rolling down a beach, leaves rustling in the wind, a book thudding on a
desk, or a plate crashing on the floor all make sounds, but these sounds are
not music. Music is sound that's organized by people on purpose, to dance
to, to tell a story, to make other people feel a certain way, or just to sound
pretty or be entertaining.

Music is organized on many different levels. Sounds can be arranged into
notes, rhythms, textures and phrases. Melodies can be organized into
anything from a simple song to a complex symphony. Beats, measures,
cadences, and form all help to keep the music organized and
understandable. But the most basic way that music is organized is by
arranging the actual sound waves themselves so that the sounds are
interesting and pleasant and go well together.

A rhythmic, organized set of thuds and crashes is perfectly good music -
think of your favorite drum solo - but many musical instruments are
designed specifically to produce the regular, evenly spaced waves that we
hear as particular pitches (musical notes). Crashes, thuds, and bangs are
loud, short jumbles of lots of different wavelengths. The sound of surf,
rustling leaves, or bubbles in a fish tank are also white noise, the term that
scientists and engineers use for sounds that are mixtures of all the different
wavelengths (just as white light is made of all the different wavelengths, or
colors, of light).

Noise

‘on NI\I\I

A tone (the kind of sound you might call a musical note) is a specific kind
of sound. The vibrations that cause it are very regular - all the same size and
same distance apart. Musicians have terms that they use to describe tones.
But this kind of (very regular) wave is useful for things other than music, so
scientists and engineers also have terms that describe tonal sound waves. It
can be very useful to know both the scientific and the musical terms and
how they are related to each other.

For example, the closer together the waves of a tonal sound are, the higher
the note sounds. Musicians talk about the pitch of the sound, or name
specific notes, or talk about tuning. Scientists and engineers, on the other
hand, talk about the frequency and the wavelength of the sound. They are
all essentially talking about the same thing. The scientific terms aren't
necessary for the musician, but they can be very helpful in understanding
and talking about what's happening when people make music.

The Concepts and Where to Find Them

¢ Wavelength - An introduction to wavelength, frequency, and pitch is
presented in Frequency, Wavelength, and Pitch. You can find out more
about the (Western) musical concept of pitch in Pitch: Sharp, Flat, and
Natural Notes.

¢ Wave Size - The other measurement you can make of regular, tonal
waves is the size of each individual wave - its "height" or "intensity"
rather than its wavelength. In sound waves, this is a measurement of
the loudness of the sound. Amplitude is a short discussion of wave
size. Musicians have many terms to discuss what they call Dynamics.

¢ Types of Waves - There are two basic types of waves. Most diagrams
show transverse waves which "wave" up-and-down as they move left-
and-right. These are easier to show in a diagram, and most of the
familiar kinds of waves - light waves, radio waves, water waves - are

transverse. But sound is made of longitudinal waves, which "wave" in
the same direction that they move. These are harder to draw, and a
little harder to imagine, than transverse waves, but you will find some
helpful suggestions at Transverse and Longitudinal Waves.

e Standing Waves - Most natural sounds are not tones. In order to
produce the extremely regular vibrations that make tonal sound waves,
musical instruments, se3e Standing Waves and Musical Instruments
and Standing Waves and Wind Instruments. To find out more about
how the waves created in an instrument are related to each other
musically, see Harmonic Series and ‘Tuning Systems.

e Sound and Ears - For a brief description of what happens when a
sound reaches your ear, see Sound and Ears

e The Math - Students struggling with the math needed for these ideas
can look at Musical Intervals, Frequency and Ratio and Powers, Roots,
and Equal Temperament.

Suggestions for Presenting These Concepts in the Classroom

¢ Decide which of the concepts you will be presenting to your class, and
prepare your lectures/presentations accordingly. You will probably
need about one class period for each related set of concepts. Sound and
Ears is particularly geared towards younger students. The concepts in
Frequency, Wavelength, and Pitch, Transverse and Longitudinal
Waves, and Amplitude can be presented to just about any age.
Standing Waves and Musical Instruments, Standing Waves and Wind
Instruments, Harmonic Series and Tuning Systems are probably best

presented to older students (middle school and up). Musical Intervals,

be used either to remind older students of the math that they have
learned and its relevance to music, or as extra information for younger
students working on these math concepts.

e Include suggested activities, worksheets, and demonstrations whenever
possible, particularly for younger students.

e Younger students will benefit from the activities and worksheets in
Sound and Music.

e¢ Worksheets that cover the basic concepts for older students are
available here. Download and copy these PDF files as handouts for

your class: Sound Waves handout and Waves Worksheet. There is also
a Worksheet Answer Key. In case you have any trouble with the PDF
files, these handouts are also included as figures at the end of this
module, but they will look better if you print out the PDF files.

e Use the exercises in the modules for class participation and discussion.

Sound Waves

Longitudinal and Transverse Waves

In longitudinal waves, the

waves “pile up” In the same direction that they are m
Sound waves are longitudinal waves,

but they are often pictured as If they were transverse,
because its easier to picture.

All waves are moving from left to o>

Transverse
High Waves pile up: up and down

In transverse waves, the

waves "pile up” in a different direction
from the direction that they are moving.
Light waves and water waves are transverse waves. Low

Short wavelength means lots of wawes
high frequency, high sournd
Frequency —

a7 eV AV AV AC AV AC AU AN,
The longer the wavelength, - C
the lower the frequency, — ee ee ee ¢
and the lower the sound. ~

Long wavelength means fewer wanes;
low frequency, low sound

Amplitude

The bigger the difference
in the highs and lows of the waves,
the louder the sound.

Louder

Te eS ee ae ee

Sound and Music Worksheet

Match both the science/engineering terms on the left and the music terms on the right
with the definitions in the middle. You will use some of the definiitons twice.

A. Waves in the air caused by vibrations

_. Low Frequency B, Waves that move in one direction,
but "wave" in another direction
____ Longitudinal Waves
C. Waves that move and "wave" in the same direction
____ Frequency ____ Low note
D. The distance between one wave and the next wave
_. High Amplitude _. Piteh
E. How often a single wave goes by
____ White Noise ___. Dynamic level
F, How big the difference is between the high points
Amplitude and the low points of the waves —_— Soft note
___. Sound Waves G. Big difference between highs and lows —. Music
___ Standing Waves H. Small difference between highs and lows __— High note
Transverse Waves 1, Lots of short waves —— Sounds
_. Wavelength J. Very few long waves —.. Loud note
____ High Frequency K. Waves that can keep vibrating in or on something
for a long time. because they "fit"
Low Amplitude
L. A sound that is a mixture of all wavelengths
M. Sounds that are organized by people
Give short answers:

1, Can sound travel through empty space? Why or why not?

2. How are sound waves like water waves? How are they not like water waves?

3. Name 2 ways a player of a musical instrument can change the sound of the instrument.

4. How can an instrument with only 4 strings get more than 4 different pitches?

5. When a trumpet player pushes down a valve, she opens an extra loop of tubing.
What does this do to the trumpet? To the sound?

Sound and Music Worksheet

Match both the science/engineering terms on the left and the music terms on the right
with the definitions in the middle. You will use some of the definiitons twice.

A. Waves in the air caused by vibrations

J Low Frequency B. Waves that move in one direction,
but “wave” in another direction
C_ Longitudinal Waves

C. Waves that move and "wave" in the same direction
E_ Frequency J Low note
D. The distance between one wave and the next wave
_G@ High Amplitude _E Pitch
E. How often a single wave goes by
L_ White Noise _F Dynamic level
F. How big the difference is between the high points
_F Amplitude and the low points of the waves _H_ Soft note
_A Sound Waves G. Big difference between highs and lows _™_ Music
_* Standing Waves H, Small difference between highs and lows _! High note
B Transverse Waves 1. Lots of short waves _A sounds
_P_ Wavelength J. Very few long waves _& Loud note

J High Frequency K. Waves that can keep vibrating in or on something
for a long time, because they "fit"
_#_ Low Amplitude
L,. A sound that is a mixture of all wavelengths

M. Sounds that are organized by people
Give short answers:

1. Can sound travel through empty space? Why or why not?
No; there can be no sound vibrations where there is no air.

2. How are sound waves like water waves? How are they not like water waves?
Both can have frequency and amplitude, but water waves are transverse
and sound waves are longitudinal.
3. Name 2 ways a player of a musical instrument can change the sound of the instrument.

They can make the pitch higher or lower or make the sound louder or softer.
4, How can an instrument with only 4 strings get more than 4 different pitches?
You cam make the vibrating part of the string shorter, ond the piteh
by holding the string down with one finger.
5. When a trumpet player pushes down a valve, she opens an extra loop of tubing.
What does this do to the trumpet? To the sound?

This in effect makes the trumpet longer, so the sound is lower.

Transverse and Longitudinal Waves

For middle school and up, a short explanation of the difference between
transverse and longitudinal waves, with some suggestions for classroom
presentation.

Waves are disturbances; they are changes in something - the surface of the
ocean, the air, electromagnetic fields. Normally, these changes are travelling
(except for Standing Waves); the disturbance is moving away from
whatever created it.

Most kinds of waves are transverse waves. In a transverse wave, as the
wave is moving in one direction, it is creating a disturbance in a different
direction. The most familiar example of this is waves on the surface of
water. As the wave travels in one direction - say south - it is creating an up-
and-down (not north-and-south) motion on the water's surface. This kind of
wave is very easy to draw; a line going from left-to-right has up-and-down
wiggles. So most diagrams of waves - even of sound waves - are pictures of
transverse waves.

But sound waves are not transverse. Sound waves are longitudinal waves.
If sound waves are moving south, the disturbance that they are creating is
making the air molecules vibrate north-and-south (not east-and-west, or up-
and-down. This is very difficult to show clearly in a diagram, so most
diagrams, even diagrams of sound waves, show transverse waves.

Note:It's particularly hard to show amplitude in longitudinal waves. Sound
waves definitely have amplitude; the louder the sound, the greater the
tendency of the air molecules to be in the "high" points of the waves, rather
than in between the waves. But it's easier show exactly how intense or
dense a particular wave is using transverse waves.

Longitudinal waves may also be a little difficult to imagine, because there
aren't any examples that we can see in everyday life. A mathematical
description might be that in longitudinal waves, the waves (the

disturbances) are along the same axis as the direction of motion of the
wave; transverse waves are at right angles to the direction of motion of the
wave. If this doesn't help, try imagining yourself as one of the particles that
the wave is disturbing (a water drop on the surface of the ocean, or an air
molecule). As it comes from behind you, a transverse waves lifts you up
and then drops you down; a longitudinal wave coming from behind pushes
you forward and then pulls you back. You can view animations of
transverse wave or by_a longitudinal wave, and particles being disturbed by
transverse and longitudinal waves.

Transverse and Longitudinal Waves
— > Allwaves are moving left-to-right —_

Longitudinal Waves
Waves "pile up" left to right

Transverse Waves
Waves "pile up" up-and-down

In water waves and other transverse
waves, the ups and downs are in a different
direction from their forward movement.
The highs and lows of sound waves and
other longitudinal waves are arranged in
the "forward" direction.

Presenting These Concepts in a Classroom

Watching movies or animations of different types of waves can help
younger students understand the difference between transverse and
longitudinal waves. The handouts and worksheets at Talking about Sound
and Music include transverse and longitudinal waves. Here are some
classroom demonstrations you can also use.

Waves in Students

Procedure

8

You will not need any materials or preparation for this demonstration,
except that you will need some room.

. Have most of the students stand in a row at one side of the classroom,

facing out into the classroom. Let some of the students stand across the
room from the line so that they can see the "waves".

. Starting at one end of the line, have the students do a traditional

stadium "wave". If they don't know how, have them all start slightly
bent forward with hands on knees. Explain that the student on the end
will lift both arms all the way over their heads and then put both down
again. Each student should do the same motion as soon as (but not
before) they feel the student beside them do it.

. If they do it well, the students watching should see a definite

transverse wave travelling down the line of students.

. Starting with the same end student, next have the line make a

longitudinal wave. Have the students start with their arms out straight
in front of them. As the wave goes by, each student will swing both
arms first toward, and then away, from the next student in line.

. Let the students take turns being the first in line, being in line, and

watching the line from the other side of the room. Let them experiment
with different motions: hopping in place, swaying to the left and right,
taking a little step down the line and back, doing a kneebend, etc.
Which kind of wave does each motion create?

Jumpropes and Slinkies

Materials and Preparation

Rope - A jump-rope is ideal, or any rope of similar weight and
suppleness

Coil - A Slinky toy works, or any metal or plastic coil with enough
length and elasticity to support a visible longitudinal wave

¢ Pole - A broomstick is fine, or a dowel, rod, pipe, or any long, thin,
rigid, smooth cylinder.

e You may want to practice with these items before the demonstration, to
make certain that you can produce visible traveling waves.

Procedure

1. Load the slinky onto the broomstick and stretch it out a bit. Have two
people holding the broomstick horizontally at waist level, as steadily
as possible, or secure the ends of the broomstick on desks or chairs.

2. Holding one end of the slinky still, have someone jerk the other end of
the slinky forward and back along the broomstick as quickly as
possible. This should create a longitudinal wave that travels down the
slinky to the other end. (If the other end is being held very tightly, but
without interfering with its coils, you may even be able to see the wave
reflect and travel back up the slinky.)

3. Secure or have someone hold one end of the jumprope very still at
waist height. Stretch the jumprope out taut, horizontally.

4. Have the person at the other end of the jumprope suddenly jerk the end
of the rope up and down again. You should see a transverse wave
travel to the other end of the rope. If the other end is secured very
tightly, you may even be able to see a reflection of the wave travel
back to the other end.

5. With both of these setups, you can experiment with sending single
pulses, multiple waves, or even try to set up standing waves. In fact, a
jumprope is usually used to make a sort of three-dimensional standing
wave of the fundamental of the rope length. Try making the standing
wave in two dimensions, going just up-and-down (without the forward
and back part of the motion). With a good rope and some practice, you
may be able to get a second harmonic standing wave, with one side of
the rope going up while the other side goes down, and a node in the
middle of the rope.

Frequency, Wavelength, and Pitch
For middle school to adult, an explanation of the relationships between
frequency, wavelength, and pitch.

Any sound that you hear as a tone is made of regular, evenly spaced waves
of air molecules. The most noticeable difference between various tonal
sounds is that some sound higher or lower than others. These differences in
the pitch of the sound are caused by different spacing in the waves; the
closer together the waves are, the higher the tone sounds. The spacing of the
waves - the distance from the high point of one wave to the next one - is the
wavelength.

All sound waves are travelling at about the same speed - the speed of sound.
So waves with a longer wavelength don't arrive (at your ear, for example) as
often (frequently) as the shorter waves. This aspect of a sound - how often a
wave peak goes by, is called frequency by scientists and engineers. They
measure it in hertz, which is how many wave peaks go by in one second.
People can hear sounds that range from about 20 to about 17,000 hertz.

The word that musicians use for frequency is pitch. The shorter the
wavelength, the higher the frequency, and the higher the pitch, of the sound.
In other words, short waves sound high; long waves sound low. Instead of
measuring frequencies, musicians name the pitches that they use most
often. They might call a note "middle C" or "2 line G" or "the F sharp in the
bass clef". (See Octaves and Diatonic Music and Tuning Systems for more
on naming specific frequencies.) These notes do have definite frequencies
(Have you heard of the "A 440" that is used as a tuning note?), but
musicians usually find it easier just to use the note names.

Wavelength, Frequency, and Pitch

The waves are all travelling at about the same speed,
so this is the number of each wave that will reach the ear
ina hundredth of a second.

\ — — —
——
Short wavelength means lots of waves; Ps
\ high frequency, high sound
(esesensennemnatel

Long wavelength means fewer waves;
low frequency, low sound

Since the sounds are travelling
at about the same speed, the one
with the shorter wavelength will

go by more frequently; it has a

higher frequency, or pitch. In
other words, it sounds higher.

Ideas for Introducing These Concepts in the Classroom

e For younger students, the "Strings Instruments" and "Wind
Instruments" activities in Sound and Music give children a chance to
create higher and lower pitched sounds. There are also handouts and
worksheets for younger students covering basic acoustics terms,
including frequency and wavelength.

e For older students, there are more advanced handouts and worksheets
in Talking about Sound and Music that cover acoustics concepts,
including frequency, wavelength, and pitch.

e If it can be arranged, a demonstration with a real musical instrument
(or two) should be popular. A live show-and-tell-style demonstration
would be most memorable, although a video or a recording with
pictures will do. Include a discussion on why and how instruments
produce higher and lower sounds. Have the musician demonstrate low
and high notes, and explain and demonstrate how the sounding part of
the instrument is being made shorter or longer to get different notes.
Point out that smaller, shorter instruments make shorter waves and
higher sounds, and larger, longer instruments make longer waves and
lower sounds. Ask the students if they are listening to a small, high-
sounding instrument, or a large, low-sounding one.

Sound Amplitude and Musical Dynamics

The amplitude of the sound waves are described in music as dynamic
levels. Unlike scientific measures of amplitude, dynamics describe the
perceived loudness of the music.

When sound waves come as a very regular, pitched tone, there are two
useful measurements you can make that tell you something about both the
sound waves and about the tone they are making. One measurement is the
distance between one wave and the next. This is the wavelength, which is
also related to the frequency and the pitch of the sound. The other
measurement you can make is the size of each individual wave - its "height"
or "intensity" rather than its length. This is the amplitude of the wave, and
it determines the loudness of the sound.

Wavelength and Amplitude

Wavelength

C y Amplitude

PPPDL TP

The wavelength is the distance between the
"crests" of two waves that are next to each other.
The amplitude is how high the crests are.

You may want to note that sound waves are not the type of waves shown in
the figure above. (Please see Transverse and Longitudinal Waves for more
on this.) Rather than piling up high in the crests of the waves, as water on
the surface of the ocean does, the air molecules in sound waves pile into the
waves. So the bigger the amplitude of the wave, the more air molecules are
in the "crest" of each wave, and the fewer air molecules are left in the "low"
spots. The amplitude of the wave is still measuring the same thing - how
much change there is during one wave - but this is more difficult to show
clearly in a diagram with sound-type longitudinal Waves waves.

Higher
Amplitude '\ I \ J \JIV\JS\y

Lower
Amplitude li i i a ae

; ede. a ae ecto, “fe, eae,
corte . o oO eA =: : . .} .
Lower eve, _ eee. ee, tee e eee
mplitude . yl o? <*, °°. ee ° a ° =
Amplitud . he to et a rx

It's easier to spot differences in amplitude at a
glance when figures use transverse waves.

Engineers and scientists call how big a wave is its amplitude. They
measure the amplitude of sound waves in decibels. Leaves rustling in the
wind are about 10 decibels; a jet engine is about 120 decibels.

Musicians call the loudness of a note its dynamic level. Forte (pronounced
"FOR-tay") is a dynamic level meaning "loud"; piano is a dynamic level
meaning "soft". Dynamic levels don't correspond to a measured decibel
level. For example, an orchestra playing "fortissimo" (which basically
means "even louder than forte") sounds much louder than a string quartet
playing "fortissimo". (See Dynamics for more of the terms that musicians
use to talk about loudness.)

Amplitude is Loudness

Louder

_ SA vd OF VF

The size of a wave (how much it is "piled up" at
the high points) is its amplitude. For sound

waves, the bigger the amplitude, the louder the
sound.

Sound and Music Activities

Lesson plans for investigative activities, appropriate for grades 3-6, that
introduce the physics of sound and music, and that explore the ways
musical instruments make sounds.

Introduction

Different musical instruments produce sounds in very different ways, but all
of them take advantage of some of the fundamental properties of sound -
the physics of sound - to make a variety of interesting and pleasant sounds.
Activity, and Resonance Activity, as well as worksheets appropriate for
younger students. All of these explore some basic concepts of sound wave
physics (acoustics) while demonstrating how various musical instruments
produce sounds.

Goals and Standards

¢ Goals - The student will develop an understanding of the physical
(scientific) causes of musical sounds, and be able to use appropriate
scientific and/or musical terminology to discuss the variety of possible
musical sounds.

e Music Standards Addressed - National Standards for Music
Education standard 8 (understanding relationships between music, the
other arts, and disciplines outside the arts)

e Other Subjects Addressed - In encouraging active exploration of the
effects of physics on music and musical instruments, these activities
also address National Science Education Standards in physical science
and in science and technology.

¢ Grade Level - 3-8

e Student Prerequisites - If younger students are not ready to conduct
their own lightly-supervised investigations, these activities should be
done as full-classroom demonstrations.

e Teacher Expertise - Teacher expertise in music is not necessary to
present this activity. The teacher should be familiar and comfortable
with basic acoustics terms and concepts (see Acoustics for Music
Theory).

e Time Requirements - Reserve one (approximately 45-minute) class
period for each activity/discussion, and one class period to finish
discussions, draw conclusions, do worksheets, and reinforce terms and
concepts. If you have a longer period of time and a large area to work
in, you may want to set up each experiment as a "work station" and
have student groups move from one station to another.

You can do any one or any combination of the activities. While doing them,
introduce whichever of the terms and concepts you think will benefit your
students. You can either use only the scientific terms, or only the musical
terms, or both. To reinforce the concepts and terms with younger students,
follow the activities with the worksheets in the Terms and Concepts section
below. For older students, present the relevant information from Frequency,
Longitudinal Waves, and include the worksheet and handout from Talking
About Sound and Music.

Terms and Concepts

During or after your activities, introduce the following terms and concepts
to the students. Worksheets to help you do this with younger students are
available here as PDF files: Terms Worksheet, Matching Worksheet,
Answer sheet. (Or you may copy the figures.) With younger students, you
may also want to study Sound and Ears. For older students, use the
worksheet and handout in Talking About Sound and Music. For more
detailed information on this subject, you may also see Talking about Sound

Transverse and Longitudinal Waves, Standing Waves and Musical
Instruments, Standing Waves and Wind Instruments, or Acoustics for Music
Theory. Use the discussion questions during and after the activities to help
the students reach conclusions about their investigations.

Terms and Concepts

e Sound Waves - When something vibrates, it makes a sound. The
vibrations travel out in all directions from the "something" in the same
way that ripples travel out from a pebble that has been dropped in
water. But instead of being waves of water, these are waves of

vibrations of air: sound waves. Because it is the air itself that is
vibrating, sound waves, unlike water waves, are invisible.

Frequency - or Pitch - Think of water waves again. They can be close
together or far apart. If they are close together, there are more of them;
they are more frequent. Frequency is the term that scientists and
engineers use to describe how many pulses of a sound wave arrive at
your ear in one second. Musicians use the term pitch. A sound with a
higher frequency (more waves) has a higher pitch, and sounds higher.
Amplitude - or Dynamic Level - Water waves can also be great, big,
tall waves, or small ripples. The size of a wave is called its amplitude.
In sound waves, the bigger the wave, the louder the sound is.
Musicians call the loudness of a sound its dynamic level.

Differences in Waves

This ear-shaped boat is not moving downstream with the waves because it is tied to the dock.

It only moves up and down as the waves arrive, as your cardrum vibrates whenever a sound wave arrives.
Which waves are like soft sounds? Loud sounds?

Which waves are like low sounds? High sounds?

Waves can be short and close together...

or they can be long and far apart.

So

Waves can be small, with not much difference
between the low points and high points...

Or they can be piled up high.

Match the Sound Waves

Which size instrument will make a high sound? A low sound? Which waves show a high sound?
Which waves show a louder sound and which show a softer one?
Draw a line from each instrument or group of instruments to the correct sound wave.

Answers

\ a

Teer Terres High Sound
ha a a Low Sound

Loud Sound

Strings Activity
Objectives and Assessment

e Objectives - The student will construct a simplified version of a
stringed instrument, using rubber bands as strings, and will use the
instrument to explore the effects of various string characteristics on
frequency and amplitude.

¢ Evaluation - Assess student learning using worksheets or answers to
discussion questions.

Materials and Preparation

Most students will be able to do this experiment alone or in small
groups. If you do not want students working with thumbtacks, plan to
use boxes or pans as instrument bodies.

You will need lots of rubber bands, as many different lengths and
thicknesses and tightnesses as you can find. If you are using boxes, the
rubber bands must be long enough to stretch around a box.

You will also need either small, sturdy cardboard or plastic boxes or
containers, with or without lids, OR pieces (about 8" X 10" or so) of
thick, flat cardboard, OR square or rectangular baking pans, one for
each student or group.

If you are using flat cardboard, you will also need thumbtacks or push
pins.

If you are using a lidded box, pencils, pens, or other objects
approximately the size and shape of a pencil (a couple for each
instrument) will be useful.

You may want scissors that are strong enough to cut the cardboard or
plastic.

If a stringed-instrument player is available for a show-and-tell, you
may want to include this after the activity, to demonstrate and
reinforce some of the main points. Any stringed instrument (guitar,
violin, harp, etc.) will do.

For older or more independent students, you may want to make copies
of the discussion questions.

Procedure

1. Each student or group should choose a variety of rubber bands (3-6,
depending on the size of their "instruments") to start with.

2. If you are using flat cardboard, stretch each rubber band between two
thumbtacks so that it is tight enough to give a particular pitch.

3. If using a box or baking pan, stretch the rubber bands around the box
or pan.

4. Have the students pluck each rubber band separately and listen
carefully to the "twang". They are listening for which ones sound
higher and which sound lower.

5. To try many different thicknesses and tightnesses, students can trade
rubber bands with each other or trade off from the central pile if there
are enough.

6. Students with the thumbtack instruments can vary length and tightness
by changing the distance between the thumbtacks.

7. Students with box or pan instruments can vary tightness by pulling on
the rubber band at the side of the box while plucking it at the top.
Students with lidded box instruments can vary length by slipping a
pencil under each end of the rubber bands on the top of the box and
then varying the distance between the pencils, or even holding the
rubber band down tightly with a finger between the pencils, in the
same way as a real string player.

8. Students with box instruments can also see if the body of the
instrument makes any difference to the sound. Can they play the
instrument with the lid off and with it on? Does cutting a hole in the lid
change the sound? Does it make it easier to play? Does adding the
pencils change the sound or make it easier to play? Do different boxes
make a different sound with the same rubber bands? Do cardboard
boxes sound different from plastic ones?

9. Ask younger students the discussion questions while they are
experimenting. Allow them to check and answer immediately.
Summarize the answers for them on the board, or remind them and let
them write them down when they are done experimenting. Give older
students a list of the discussion questions before they begin.

Discussion Questions

¢ Do thicker rubber band "strings" sound higher or lower than thinner
ones? (Answer: thicker should sound lower.)

e Do tighter strings sound higher or lower than looser ones? (Tighter
should sound higher.)

¢ Do shorter strings sound higher or lower than longer ones? (Shorter
should sound higher.)

e Do there seem to be differences in how loud and soft or how dull or
clear a string sounds? If so, what seems to cause those differences?

e What determines whether the sound of a string is loud or soft?

e What happens to the sound if they pluck with one finger while
touching the string lightly with another finger? (No "twang"; the touch
stops the vibrations.) If their instrument design allows it, what happens
when they hold the string tightly down against the instrument and then
pluck it? (The shorter vibrating length should give a higher pitch.)

e After their experiments, can they explain what happens when a player
holds a string down with a finger? What if the same string is held
down in a different spot?

e Based on their observations, do the students feel they could tell which
strings of an instrument are the low strings just by looking at them
closely? (For an extra activity, arrange for them to try this with a real
instrument.)

e Can the students come up with possible reasons why the thickness,
length, and tightness of a string affect its frequency/pitch in the way
that they do? (For example, why does a shorter string have a higher
frequency/pitch?) (It may help on length to remind them that the
longer the waves are, the less frequent they will be.)

Wind Instruments Activity
Objectives and Assessment

e Objectives - The student will explore the effects of air column size
(and shape) on the frequency and amplitude of standing waves in the
air column, using empty glass bottles, and water if necessary to vary
air column size.

e Evaluation - Assess student learning using worksheets or answers to
discussion questions.

Materials and Preparation

If you do not want your students working with glass jugs and water,
plan to do this as a demonstration.

You will need several narrow-necked bottles, all the same size and
shape OR several narrow-necked bottles of varying sizes and shapes.
Bottles should be empty and clean. Make sure before the class begins
that your bottles give a clear, reasonably loud sound when you blow
across the top of them. If necessary, practice getting a sound. Large
glass jugs with an inner lip diameter of approximately one inch work
well.

If using bottles of the same size, you will also need water to fill them
to varying depths. If you are using this approach, food coloring is very
useful to clearly show the depth of the water.

If plastic recorders are available to your students, or a player of a
woodwind or brass instrument is available for a show-and-tell, they
can be used for an extra demonstration.

For older or more independent students, you may want to make copies
of the discussion questions.

Procedure

1. If using same-size bottles and water, fill each bottle to a different depth

(for example, an inch in one bottle, two inches in another, three inches
in a third and so on). If you have food coloring, add a few drops to the
water in each bottle so it is easy to see the depths.

2. Make the air in a bottle vibrate by blowing steadily across the top of

the bottle.

3. "Play" each bottle in turn, and arrange them in order from the highest
sound to the lowest.

4. If you have the time and inclination, you can even try to "tune" the
bottles by adding or pouring out water.

5. If recorders or a wind instrument are available, demonstrate how
covering and uncovering the holes on the instrument changes the pitch.
Explain that the main vibration in the instrument is happening in the
air inside the instrument (just like the air in the bottles), in between the
mouthpiece and the first hole that the air can escape from..

Discussion Questions

e If using bottles of different shapes and sizes, how does the size of the
bottle affect the pitch/frequency? Does the shape of the bottle seem to
affect it?

¢ Does the size and shape of the bottle seem to affect anything else, like
the loudness of the sound or the tone quality?

e What do you think explains these effects?

e If using water in bottles, how does the amount of water affect the
pitch/frequency? Why? (You may need to remind the students that it is
the air in the bottle that is vibrating; more water means a smaller space
for the air; smaller space means shorter waves and higher
frequency/pitch).

e How is a bottle "instrument" the same as a wind instrument, and how
is it different?

e If demonstrating with instruments: How does opening and closing the
holes of the instruments change the pitch? Why? (Answer: the shorter
the distance between the mouthpiece and the first open hole, the
shorter the waves and the higher the pitch/frequency. Opening and
closing other holes further down the instrument from the first open
hole may have no discernible effect - they are not changing the length
of the vibrating column of air - or if they are affecting the vibrating air
a little, they may change the sound enough to make it more or less in
tune.) If a brass instrument is used, what is the effect of opening a
valve or extending the slide? (Opening valves actually lengthens the
instrument, by opening up extra tubing, lowering the pitch.)

Percussion Activity
Objectives and Assessment

¢ Objectives - The student will assist in constructing a "found objects"
chime, and will use the instrument to explore the effects of various
object characteristics on frequency and amplitude.

e Evaluation - Assess student learning using worksheets or answers to
discussion questions.

Materials and Preparation

e Each working group will need a dowel, rod, or small beam, around 4-6
feet long, held at both ends about five feet off the ground.

e Each group will need a variety of objects of different sizes and
materials. Forks, spoons, spatulas, rulers, wind chimes, lengths of
chain, lengths of pipe or bamboo or tubing, are all easy to line up
below the dowel because they are long and thin. Objects that have
holes or handles (slotted spoons, pan lids) making it easier to keep
them tied on, are also a good idea. Objects that are metal, hardwood,
hard plastic, hollow, and/or made in a single piece are most likely to
make easy-to-hear, interesting sounds.

e It may be easier to answer some of the discussion question if some of
the objects are similar objects in a variety of sizes, for example small
medium and large metal spoons.

e You will need enough string to hang the objects from the dowels, and
may need tape to keep the objects on the string. Keep in mind, though,
that tape will probably dampen the vibrations of the object so that it
won't "ring" as long.

e You will need something the students can use to strike the objects; a
wooden spoon, short stick, pen or pencil, or ruler. Or they can
experiment with using different objects as "drumsticks". Which do the
students prefer and why?

e For older or more independent students, you may want to make copies
of the discussion questions.

Procedure

1. Have the students hang the objects securely from the dowel.
2. The students should then strike the objects one at a time, listening

carefully to the sound each object typically makes.

Discussion Questions

Does the size of the object seem to affect its pitch/frequency? Its
loudness?

Does the shape of the object seem to affect its pitch/frequency? Its
loudness?

Does the object's material seem to affect its pitch/frequency? Its
loudness?

Can you tell what effects the thickness of an object has on its sound?
What seems to affect how long a sound lasts?

What objects make the sounds that you like best? Which do you think
would make good percussion instruments? Why?

Which of these effects do you think you can explain in terms of waves
and the vibrations the objects must be making?

Instrument Body Activities

Objectives and Assessment

Objectives - The student will construct a simple megaphone, and will
use the megaphone and a music box in several simple investigations to
explore the effects that the body of an instrument has on its sound.
Evaluation - Assess student learning using worksheets or answers to
discussion questions.

Materials and Preparation

Decide whether each step of this investigation will be a teacher
demonstration or an individual or small-group activity.

You will need a music box.

You will need several large, flat surfaces of different types of materials
- different types of wood and metal as well as plastic and softer
surfaces will be particularly instructive. A box or drawer made of
hardwood is optional.

You will also need large sheets of paper, construction paper,
newspaper, soft, pliable plastic or foam or poster board, and some tape,
OR a megaphone. If you have a variety of megaphone materials, have
different students use different materials to see if material choice
affects the sound.

For older or more independent students, you may want to make copies
of the discussion questions.

Procedure

1,

Z

Wind the music box and let everyone listen to it while holding it in
your hand.

Place the box on different surfaces and listen to the difference it makes
in the sound. Continue to wind it as necessary to hear a long example
of each surface. If you can, place the music box inside a wooden box
or drawer.

. If you do not have a real megaphone to demonstrate, let the students

make their own megaphones by rolling the paper into a cone shape,
open at both ends. Tape it if necessary to hold the shape.

. Let them talk or sing into their megaphones and otherwise experiment

with how the megaphone changes sounds. Experiment with different
megaphone sizes and shapes (narrow or widely flaring).

Discussion Questions

What effect does each surface have on the sound from the music box?
What is causing these effects? (Answer: some surfaces will vibrate
with the music box if they are touching. See Resonance.)

Why do instruments have bodies; why aren't they just a bunch of
strings or a reed or a membrane to beat on?

Why would an instrument maker choose to make an instrument body
out of wood (like a violin or piano)? Why might metal be chosen (as in
brass and many percussion instruments)? Of the other materials you
experimented with, would you make instruments out of them? What
kind of instrument with each material? Why?

How does a megaphone shape change a sound? Does it matter whether
the megaphone is narrow or flaring?

How do you think the megaphones would have changed if they had
been made of wood or of metal?

Would a violin sound louder if you were sitting right in front of it or
next to it? What about a trumpet? What's the difference?

Based on your observations, what do you think the shape of the
instrument does to the sound of a tuba, trumpet, trombone, clarinet, or
saxophone? What about flutes and bassoons (which do not flare)?

Sound and Ears
For middle school and up, an introduction to the human ear.

Introduction

The ear is the sense organ that picks up sound waves from the surrounding
air and turns them into nerve impulses that can be sent to the brain. The
sound waves carry lots of information - language, music, and noises - all
mixed up together. The task of the ear is to turn the signals in these waves
of bouncing air molecules into electrical nerve signals, while keeping as
much of the information in the signal as possible. (Then it's the brain's job
to sort the signals and make sense out of them.) It's not easy to turn one
kind of signal into another kind without losing information, but the ear is
well designed for the task.

Note: The human ear also has some other functions not related to hearing;
those won't be discussed here.

When something vibrates, the vibrations can travel as waves through solids,
liquids, and gases. Even animals that have no ears can often feel these
vibrations. But in order to understand language and hear music, the brain
has to be given more information than just "there's a vibration". It needs to
know the frequency and amplitude of all the waves that the ear is collecting.
Interestingly, the ear sends this information to the brain very accurately by
turning the sound waves in the air (vibrations in a gas) into vibrations in
bones (solid), and then into waves in a fluid in the inner ear (a liquid),
before they become (electrical) nerve signals. This might seem like a lot of
unnecessary translation, but it allows the sense of hearing to be both sturdy
and very sensitive, as explained below.

Parts of the Ear

The ear has three main sections. In the outer ear, the sound waves are still
moving in air. In the middle ear, the sound waves are being conducted by
three small bones. In the inner ear, the waves are moving through the fluid-
filled cochlea.

The Ear

ee

U

Inner Ear
Middle Ear
——

yo

| =a |
Hammer

Ear Canal Anvil
Stirrup

The parts of the ear that
aren't involved in hearing
have been left out.

The Outer Ear

The part of the human ear that you can see is simply a sound wave
collector. Its shape helps to funnel the sound waves into the auditory canal
(or ear canal) so that you get plenty of signals from even soft sounds,
particularly ones from the direction that you are looking at. At the other end
of the ear canal is the eardrum (or tympanic membrane). This is a
membrane that is stretched tight, like the membranes on the tops of drums
(including tympani). And thin, taut membranes are very good at vibrating,
which is why they can be found both on drums and inside ears. The
eardrum picks up the vibrations in the ear canal and vibrates with them.

The Middle Ear

On the other side of the eardrum are the three tiny bones of the middle ear,
the hammer, the anvil, and the stirrup. They are named for their shapes.
Vibrations in the eardrum are passed to the hammer, which transmits them

to the anvil, which makes the stirrup vibrate against the oval window of the
cochlea. Bone is a very good conductor of vibrations, and the bones of the
middle ear are specially arranged so that they can amplify (make louder)
very quiet sounds. On the other hand, if things get too loud, tiny muscles in
your middle ear can relax the eardrum a bit. A relaxed eardrum doesn't
vibrate as much (think of a relaxed rubber band as opposed to a taut one),
and this helps to keep things from getting damaged.

The Inner Ear

The cochlea is a fluid-filled spiral (shaped something like a snail shell)
about the size of a pea. Vibrations in the stirrup make waves in the fluid that
travel down the spiral. In the fluid, in a long strip following the spiral, is the
organ of Corti. This organ is covered with (about 20,000) tiny, incredibly
sensitive hairs that are waving around inside the Cochlear fluid. Each of
these hairs is a nerve ending that is picking up specific information about
the vibrations in the fluid. At the end of the organ of Corti, the nerves are
bundled together as the auditory nerve, which brings the information to the
brain.

Note: The fragile, sensitive hairs on the organ of Corti would never stand
up to the rough conditions in the ear canal. Even protected in the Coclear
fluid, they don't last forever, especially the ones that can sense the highest-
frequency vibrations. That is why most people begin to lose their sense of
hearing as they grow older.

Presenting this Module to Children

You can present the information above in the form of a classroom
lecture/presentation to elementary or middle school classes. Here are some
suggestions for making the presentation more interactive and engaging.

Locate a poster or large diagram of the ear to use as a visual aid.

If you don't have a poster, or if the printing on it is small, write the
names of the parts of the ear on the board as you discuss them.

Make copies of this PDF file worksheet for a class handout. Have the
students label the parts of the ear during or after your presentation.
When you discuss the outer ear, have the students make their own
simple funnels out of paper and tape. Have each student hold the small
end of the funnel up to an ear to see if it helps the ear collect sounds
even better, especially in the direction that the funnel is pointing. A
very simple version of this is to simply cup the hands behind the ears.
You may want to have a classroom discussion on why it might be
useful to have ears that "focus" on the sounds that are directly ahead of
you. If the class is also studying animals, you can bring in pictures of
various ears. Which are large and which small? Which are pointed
straight ahead? How would that be useful? Which can swivel in
different directions? How would that be useful? If they cannot come
up with any ideas, give them a hint by asking which animals are
hunters and which are hunted. You may also want to discuss animals
that pick up vibrations with parts of their bodies that are very unlike
human ears. You can even turn this into a class project by asking
students to research and report on different animals (reptiles,
elephants, and insects are particularly interesting).

For the eardrum, you can simply use rubber bands to demonstrate that
things vibrate more clearly when they are taut. Or if you want to be
more adventurous (and messy), stretch a sheet of thick cellophane or
thin rubber, leather, hide, or close-woven fabric across the opening of a
bowl, can, or small tub, and sprinkle some rice over it. Try hitting your
stretched membrane with a stick when it is relaxed, fairly taut, and
very taut. When is it best at transmitting the vibrations and making the
rice jump? Can you get it taut enough to act like an eardrum - so taut
that even a loud sound nearby (say, hitting a different can) will make it
vibrate and the rice jump?

When discussing vibrations in bone, let them talk while pressing their
fingers gently on the back of their jawbones (below the ears). They
should be able to feel the vibrations from their own speech in the bone
almost as well as when they press against their throats, where the
sounds are being produced. But they probably won't feel any vibrations

from their noses, cheeks, outer ear, or hair. You can point out that: a lot
of what you hear when you hear your own voice is coming to your ear
through your jawbone. That's why your voice sounds so different to
you when you hear a recording of it.

When discussing the Cochlea and organ of Corti, ask if the students
have seen underwater plants moving back and forth in the waves. If
you really want to be hands-on, you can get a tank of water, hang some
long thin ribbons or plant fronds in it, and let them make waves and
watch the "hairs" move.

Musical Intervals, Frequency, and Ratio
For high school and above, a discussion of the relationship of musical
intervals and frequency ratios, with examples and exercises.

In order to really understand tuning, the harmonic series, intervals, and
harmonic relationships, it is very useful to understand a little bit about the
physics of sound and to be comfortable discussing ratios, fractions, and
decimals. This lesson is a short review of some basic math concepts for
students who want to understand some of the math and physics principles
that underlie music theory.

Ratios, fractions, and decimals are basically three different ways of saying
the same thing. (So are percents, but they don't have anything to do with
music.)

Example:
If you have two apples and three oranges, that's five pieces of fruit
altogether. You can say:

e The ratio of apples to oranges is 2:3, or the ratio of oranges to apples
IS soe.

e The ratio of apples to total fruit is 2:5, or the ratio of oranges to total
fruit is 3:5.

e 2/5 of the fruit are apples, and 3/5 of the fruit are oranges.

e There are 2/3 as many apples as oranges, and 1 and 1/2 times (or 3/2)
as Many oranges as apples.

e There are 1.5 times as many oranges as apples, or there are only .67
times as many apples as oranges.

e 0.4 (Four tenths) of the fruit is apples, and 0.6 (six tenths) of the fruit
is oranges.

Note: You should be able to see where the numbers for the ratios and
fractions are coming from. If you don't understand where the decimal
numbers are coming from, remember that a fraction can be understood as

a quick way of writing a division problem. To get the decimal that equals
a fraction, divide the numerator by the denominator.

Example:
An adult is walking with a child. For every step the adult takes, the
child has to take two steps to keep up. This can be expressed as:

e The ratio of adult to child steps is 1:2, or the ratio of child to adult
steps is 2:1.

e The adult takes half as many (1/2) steps as the child, or the child takes
twice as many (2/1) steps as the adult.

e The adult takes 0.5 as many steps as the child, or the child takes 2.0
times as many steps as the adult.

Exercise:

Problem:

The factory sends shirts to the store in packages of 10. Each package
has 3 small, 3 medium, and 4 large shirts. How many different ratios,
fractions, and decimals can you write to describe this situation?

Solution:

e Ratio of small to medium is 3:3. Like fractions, ratios can be
reduced to lowest terms, so ratio of 1:1 is also correct.

e Ratio of small to large, or medium to large, is 3:4; ratio of large to
either of the others is 4:3.

e Ratio of small or medium to total is 3:10; ratio of large to total is
4:10.

e 3/10, or 0.3, of the shirts, are small; 3/10, or 0.3 of the shirts are
medium, and 4/10, or 0.4 of the shirts, are large.

e There are 3/4 as many small or medium shirts as there are large
shirts, and there are 4/3 as many large shirts as small or medium
shirts.

e If you made more ratios, fractions, and decimals by combining
various groups (say ratio of small and medium to large is 6:4, and
so on), give yourself extra credit.

What has all this got to do with music? Quite a bit, as a matter of fact. For
example, every note in standard music notation is a fraction of a beat, and
every beat is a fraction of a measure. You can explore the relationship
between fractions and rhythm in Fractions, Multiples, Beats, and Measures,
Duration and Time Signature.

The discussion here will focus on the relationship between ratio, frequency,
and musical intervals. The interval between two pitches depends on the
ratio of their frequencies. There are simple, ideal ratios as expressed in a
harmonic series, and then there is the more complex reality of equal
temperament, in which the frequency ratios are not so simple and are best
written as roots or decimals. Here is one more exercise before we go on to
discussions of music.

Exercise:

Problem:

The kind of sound waves that music is made of are a lot like the adult
and child walking along steadily in the example above. Low notes
have long wavelengths, like the long stride of an adult. Their
frequencies, like the frequency of the adult's steps, are low. High notes
have shorter wavelengths, like the small stride of the child. Their
frequencies, like the frequency of a child's steps, are higher. (See
Sound, Physics and Music for more on this.)

You have three notes, with frequencies 220, 440, and 660. (These
frequencies are in hertz, or waves per second, but that doesn't
really matter much; the ratios will be the same no matter what
units are used.)

1. Which note sounds highest, and which sounds lowest?

2. Which has the longest wavelength, and which the shortest?

3. What is the ratio of the frequencies? What is it in lowest terms?

4. How many waves of the 660 frequency are there for every wave
of the 220 frequency?

5. Use a fraction to compare the number of waves in the 440
frequency to the number of waves in the 660 frequency.

Solution:

For every one wave of
frequency 220, there are two
of 440, and 3 of 660.

1. 660 sounds the highest; 220 lowest. (440 is a "tuning A" or A
440", by the way. 220 is the A one octave lower, and 660 is the E
above A 440.)

2.220 has the longest wavelength, and 660 the shortest.

3. 220:440:660 in lowest terms is 1:2:3

4.3

5. There are only 2/3 as many waves in the 440 frequency as in the
660 frequency.

It is easy to spot simple frequency relationships, like 2:1, but what about
more complicated ratios? Remember that you are saying the ratio of one

frequency to another IS (equals) another ratio(or fraction or decimal).
This idea can be written as a simple mathematical expression. With enough
information and a little bit of algebra, you can solve this equation for any
number that you don't have.

If you remember enough algebra, you'll notice that the units for frequency
in this equation must be the same: if frequency #1 is in hertz, frequency #2
must be in hertz also. In all the examples and problems below, I am going to
assume all frequencies are in hertz (waves per second), but you can use any
frequency unit as long as they are both the same. Most musicians don't
talk about frequency much, and when they do, they rarely mention units,
but just say, for example, "A 440".

The idea
‘The ratio of frequency #1 to frequency # 2 is Y"
can be written:

frequency #1

frequency #2
Y can be another ratio written as a fraction, or it can be a decimal,
a root, or a whole number. You can use this equation, or an algebraic
rearrangement of it, to find a frequency or a ratio that you do not know.

Since musical intervals depend on frequency ratios, you can also use it
to find an interval if you know the frequencies involved.

Remember that ratios, fractions and
decimals are all just different ways of
writing the same idea. If you write the

ratio as a fraction it becomes easy to use
in simple algebra equations.

Example:

Say you would like to compare the frequencies of two sounds. Sound #1 is
630 and sound #2 is 840. If you use the expression given above and do the
division on a calculator, the answer will be a decimal. If you simply reduce
the fraction to lowest terms, or if you know the fraction that these decimals
represent, you can see that you have a simple ratio of 3:4. Notice that if
you switch the frequencies in the expression, the ratio also switches from

3:4 to 4:3. So it doesn't really matter which frequency you put on top; you

will get the right answer as long as you keep track of which frequency is
which.

frequency #2 frequency #2
frequency #2 a frequency #1 i
630 840 ae
rt aes =o $30" alee:
= 2 |
4 3

Sound waves in the real world of musical instruments often do have simple
ratios like these. (See Standing Waves and Musical Instruments for more
about this.) In fact, a vibrating string or a tube of vibrating air will generate
a whole series of waves, called a harmonic series, that have fairly simple
ratios. Musicians describe sounds in terms of pitch rather than frequency
and call the distance between two pitches (how far apart their frequencies
are) the interval between the pitches. The simple-ratio intervals between the
harmonic-series notes are called pure intervals. (The specific names of the
intervals, such as "perfect fifth" are based on music notation and traditions

rather than physics. If you need to understand interval names, please see
Interval.)

Example:
perfect minor

~ “py fourth ae major second

third = third

intervals: octave

Harmonic Series
on C

Harmonic Number . c
to use in I = , | ,
frequency ratios

You can use a harmonic series to find
frequency ratios for pure intervals. For
example, harmonics 2 and 3 are a perfect
fifth apart, so the frequency ratio of a
perfect fifth is 2:3. Harmonics 4 and 5 are a

major third apart, so the frequency ratio for

major thirds is 4:5. Harmonics 4 and 1 are

two octaves apart, so the frequency ratio of
notes two octaves apart is 4:1.

Perhaps you would like to find the frequency of a note that is a perfect fifth
higher or lower than another note. A quick look at the harmonic series here
shows you that the ratio of frequencies of a perfect fifth is 3:2.

Note:It does not matter what the actual notes are! If the ratio of the
frequencies is 3:2, the interval between the notes will be a perfect fifth.

The higher number in the ratio will be the higher-sounding note. So if you
want the frequency of the note that is a perfect fifth higher than A 440, you
use the ratio 3:2 (that is, the fraction 3/2). If you want the note that is a
perfect fifth lower than A 440, you use the ratio 2:3 (the fraction 2/3).

Perfect Fifth Higher Perfect Fifth Lower
frequency #1 2 frequency #1 3
frequency #2 0 E frequency #2 >

440 _ 2 ee on ae
frequency #2 3 frequency #2 2

3

440 x 3 = frequency #2 440 x 4 = frequency #2

660 = frequency #2 293.33 = frequency #2

frequency #2 = frequency #1 X ratio of #2 over #1

Remember that it is important to put
the ratio numbers in the right place; if
#2 is the higher frequency, then #2
must be the higher number in the
ratio, too. If you want #2 to be the
lower frequency, then #2 should be the
lower ratio number, too. Always check

your answer to make sure it makes
sense; a higher note should have a
higher frequency.

In this example, I have done the algebra for you to show that you are really
using the same equation as in example 1, just rearranged a bit. If you are
uncomfortable using algebra, use the red expression if you know the
interval but don't know one of the frequencies.

Pure intervals that are found in the physical world (such as on strings or in
brass tubes) are nice simple ratios like 2:3. But musicians in Western
musical genres typically do not use pure intervals; instead they use a tuning
system called equal temperament. (If you would like to know more about
how and why this choice was made, please read Tuning Systems.) In equal
temperament, the ratios for notes in equal temperament are based on the
twelfth root of two. (For more discussion and practice with roots and equal

evens out the intervals between the notes so that scales are more uniform,
but it makes the math less simple.

Example:

Frequency Ratios in Equal Temperament

frequency rahe Oecimel Equvalent

asopower (tothe nearest ten thousandth}
al the twellth root of 2

Unison 2 = 1.0000

Minor Second 2 = 1.0595
Majer Sead (v2!) = 1.1225
Minor Third | 20 «= ~=«21.1892
Major Third || 2 = 1.2599
Perfect Fourth |. 2 | = 1.3348
Tritone (2) = 1.4142
Perfect Fifth 2 = 1.4983
Minor Sixth 2 = 1-5874
Major Sixth | 2 = 1.6818
Minor Seventh 2 = 1.7818
Major Seventh 2)" = 1.8897
Octave ‘2 ' = 2.0000

Say you would like to compare a pure major third from the harmonic series
to a equal temperament major third.

Pure Major Third Equal Temperament Major Third
frequency #1 T 4
=z Y¥ 2 = 1.2599
frequency #2
2 = 1.2500

By comparing the ratios as decimal numbers, you can see that a pure major
third is quite a bit smaller than an equal temperament major third.

Exercise:

Problem:
A note has frequency 220. Using the pure intervals of the harmonic
series, what is the frequency of the note that is a perfect fourth higher?

What is the frequency of the note that is a major third lower?

Solution:

frequency #2 = frequency #1 X ratio of #2 over #1

Perfect Fourth Higher: Major Third Lower:
Ratio 4:3 Ratio 4:5
frequency #2= frequency #2=
220x 4= 293.33 220x 2 = 176
Exercise:
Problem:

The frequency of one note is 1333. The frequency of another note is
1121. What equal temperament interval will these two notes sound
like? (Hint: compare the frequencies, and then compare your answer to
the frequencies in the equal temperament figure above.)

Solution:
frequency #1

frequency #2

1333
li21 = 1.1891

This is very close to 1.1892,
so the interval will sound like a minor third.

Harmonic Series
The harmonic series is the key to understanding not only harmonics, but
also timbre and the basic functioning of many musical instruments.

Introduction

Have you ever wondered how a trumpet plays so many different notes with
only three valves, or how a bugle plays different notes with no valves at all?
Have you ever wondered why an oboe and a flute sound so different, even
when they're playing the same note? What is a string player doing when she
plays "harmonics"? Why do some notes sound good together while other
notes seem to clash with each other? The answers to all of these questions
will become clear with an understanding of the harmonic series.

Physics, Harmonics and Color

Most musical notes are sounds that have a particular pitch. The pitch
depends on the main frequency of the sound; the higher the frequency, and
shorter the wavelength, of the sound waves, the higher the pitch is. But
musical sounds don't have just one frequency. Sounds that have only one
frequency are not very interesting or pretty. They have no more musical
color than the beeping of a watch alarm. On the other hand, sounds that
have too many frequencies, like the sound of glass breaking or of ocean
waves crashing on a beach, may be interesting and even pleasant. But they
don't have a particular pitch, so they usually aren't considered musical
notes.

Frequency and Pitch

The higher the frequency, the
higher the note sounds.

When someone plays or sings a note, only a very particular set of
frequencies is heard. Imagine that each note that comes out of the
instrument is a smooth mixture of many different pitches. These different
pitches are called harmonics, and they are blended together so well that
you do not hear them as separate notes at all. Instead, the harmonics give
the note its color.

What is the color of a sound? Say an oboe plays a middle C. Then a flute
plays the same note at the same loudness as the oboe. It is still easy to tell
the two notes apart, because an oboe sounds different from a flute. This
difference in the sounds is the color, or timbre (pronounced "TAM-ber") of
the notes. Like a color you see, the color of a sound can be bright and bold
or deep and rich. It can be heavy, light, murky, thin, smooth, or
transparently clear. Some other words that musicians use to describe the
timbre of a sound are: reedy, brassy, piercing, mellow, thin, hollow,
focussed, breathy (pronounced BRETH-ee) or full. Listen to recordings of a
violin and a viola. Although these instruments are quite similar, the viola
has a noticeably "deeper" and the violin a noticeably "brighter" sound that
is not simply a matter of the violin playing higher notes. Now listen to the
same phrase played by an electric guitar, an acoustic guitar with twelve

steel strings and an acoustic guitar with six nylon strings. The words
musicians use to describe timbre are somewhat subjective, but most
musicians would agree with the statement that, compared with each other,
the first sound is mellow, the second bright, and the third rich.

Exercise:

Problem:

Listen to recordings of different instruments playing alone or playing
very prominently above a group. Some suggestions: an unaccompanied
violin or cello sonata, a flute, oboe, trumpet, or horn concerto, native
American flute music, classical guitar, bagpipes, steel pan drums,
panpipes, or organ. For each instrument, what "color" words would
you use to describe the timbre of each instrument? Use as many words
as you can that seem appropriate, and try to think of some that aren't
listed above. Do any of the instruments actually make you think of
specific shades of color, like fire-engine red or sky blue?

Solution:

Although trained musicians will generally agree that a particular sound
is reedy, thin, or full, there are no hard-and-fast right-and-wrong
answers to this exercise.

Where do the harmonics, and the timbre, come from? When a string
vibrates, the main pitch you hear is from the vibration of the whole string
back and forth. That is the fundamental, or first harmonic. But the string
also vibrates in halves, in thirds, fourths, and so on. Each of these fractions
also produces a harmonic. The string vibrating in halves produces the
second harmonic; vibrating in thirds produces the third harmonic, and so
on.

Note:This method of naming and numbering harmonics is the most
straightforward and least confusing, but there are other ways of naming
and numbering harmonics, and this can cause confusion. Some musicians
do not consider the fundamental to be a harmonic; it is just the

fundamental. In that case, the string halves will give the first harmonic, the
string thirds will give the second harmonic and so on. When the
fundamental is included in calculations, it is called the first partial, and the
rest of the harmonics are the second, third, fourth partials and so on. Also,
some musicians use the term overtones as a synonym for harmonics. For
others, however, an overtone is any frequency (not necessarily a harmonic)
that can be heard resonating with the fundamental. The sound of a gong or
cymbals will include overtones that aren't harmonics; that's why the gong's
sound doesn't seem to have as definite a pitch as the vibrating string does.
If you are uncertain what someone means by the second harmonic or by
the term overtones, ask for clarification.

Vibrating String

Whole Halves Thirds Fourths and so on...

1 2 3 4
Fundamental

The fundamental pitch is produced by
the whole string vibrating back and
forth. But the string is also vibrating in
halves, thirds, quarters, fifths, and so
on, producing harmonics. All of these
vibrations happen at the same time,
producing a rich, complex, interesting
sound.

A column of air vibrating inside a tube is different from a vibrating string,
but the column of air can also vibrate in halves, thirds, fourths, and so on, of

the fundamental, so the harmonic series will be the same. So why do
different instruments have different timbres? The difference is the relative
loudness of all the different harmonics compared to each other. When a
clarinet plays a note, perhaps the odd-numbered harmonics are strongest;
when a French horn plays the same notes, perhaps the fifth and tenth
harmonics are the strongest. This is what you hear that allows you to
recognize that it is a clarinet or horn that is playing.

Note: You will find some more extensive information on instruments and
harmonics in Standing Waves and Musical Instruments and Standing
Waves and Wind Instruments.

The Harmonic Series

A harmonic series can have any note as its fundamental, so there are many
different harmonic series. But the relationship between the frequencies of a
harmonic series is always the same. The second harmonic always has
exactly half the wavelength (and twice the frequency) of the fundamental;
the third harmonic always has exactly a third of the wavelength (and so
three times the frequency) of the fundamental, and so on. For more
discussion of wavelengths and frequencies, see Frequency, Wavelength, and
Pitch.

Harmonic Series Wavelengths and Frequencies

The second harmonic has half
the wavelength and twice the
frequency of the first. The third
harmonic has a third the
wavelength and three times the
frequency of the first. The fourth
harmonic has a quarter the
wavelength and four times the
frequency of the first, and so on.
Notice that the fourth harmonic
is also twice the frequency of the
second harmonic, and the sixth
harmonic is also twice the
frequency of the third harmonic.

Say someone plays a note, a middle C. Now someone else plays the note
that is twice the frequency of the middle C. Since this second note was
already a harmonic of the first note, the sound waves of the two notes
reinforce each other and sound good together. If the second person played
instead the note that was just a litle bit more than twice the frequency of the

first note, the harmonic series of the two notes would not fit together at all,
and the two notes would not sound as good together. There are many
combinations of notes that share some harmonics and make a pleasant
sound together. They are considered consonant. Other combinations share
fewer or no harmonics and are considered dissonant or, when they really
clash, simply "out of tune" with each other. The scales and chords of most
of the world's musics are based on these physical facts.

Note:In real music, consonance and dissonance also depend on the
standard practices of a musical tradition, especially its harmony practices,
but these are also often related to the harmonic series.

For example, a note that is twice the frequency of another note is one octave
higher than the first note. So in the figure above, the second harmonic is
one octave higher than the first; the fourth harmonic is one octave higher
than the second; and the sixth harmonic is one octave higher than the third.
Exercise:

Problem:

1. Which harmonic will be one octave higher than the fourth
harmonic?

2. Predict the next four sets of octaves in a harmonic series.

3. What is the pattern that predicts which notes of a harmonic series
will be one octave apart?

4. Notes one octave apart are given the same name. So if the first
harmonic is a "A", the second and fourth will also be A's. Name
three other harmonics that will also be A's.

Solution:

1. The eighth harmonic
2. The fifth and tenth harmonics; the sixth and twelfth harmonics;
the seventh and fourteenth harmonics; and the eighth and

sixteenth harmonics

3. The note that is one octave higher than a harmonic is also a
harmonic, and its number in the harmonic series is twice (2 X) the
number of the first note.

4. The eighth, sixteenth, and thirty-second harmonics will also be
A's.

A mathematical way to say this is "if two notes are an octave apart, the ratio
of their frequencies is two to one (2:1)". Although the notes themselves can
be any frequency, the 2:1 ratio is the same for all octaves. And all the other
intervals that musicians talk about can also be described as being particular
ratios of frequencies.

A Harmonic Series Written as Notes

e
— 5 6 7 8 9 10 11 12 #13 = «14 15 16

Take the third harmonic, for example. Its frequency is three times the first
harmonic (ratio 3:1). Remember, the frequency of the second harmonic is
two times that of the first harmonic. So the ratio of the frequencies of the
second to the third harmonics is 2:3. From the harmonic series shown
above, you can see that the interval between these two notes is a perfect
fifth. The ratio of the frequencies of all perfect fifths is 2:3.

Exercise:

Problem:

1. The interval between the fourth and sixth harmonics (frequency
ratio 4:6) is also a fifth. Can you explain this?

2. What other harmonics have an interval of a fifth?

3. Which harmonics have an interval of a fourth?

4. What is the frequency ratio for the interval of a fourth?

Solution:

1. The ratio 4:6 reduced to lowest terms is 2:3. (If you are more
comfortable with fractions than with ratios, think of all the ratios
as fractions instead. 2:3 is just two-thirds, and 4:6 is four-sixths.
Four-sixths reduces to two-thirds.)

2. Six and nine (6:9 also reduces to 2:3); eight and twelve; ten and
fifteen; and any other combination that can be reduced to 2:3
(12:18, 14:21 and so on).

3. Harmonics three and four; six and eight; nine and twelve; twelve
and sixteen; and so on.

4.3:4

Note:If you have been looking at the harmonic series above closely, you
may have noticed that some notes that are written to give the same interval
have different frequency ratios. For example, the interval between the
seventh and eighth harmonics is a major second, but so are the intervals
between 8 and 9, between 9 and 10, and between 10 and 11. But 7:8, 8:9,
9:10, and 10:11, although they are pretty close, are not exactly the same. In
fact, modern Western music uses the equal temperament tuning system,
which divides the octave into twelve notes that are spaced equally far
apart. The positive aspect of equal temperament (and the reason it is used)
is that an instrument will be equally in tune in all keys. The negative aspect
is that it means that all intervals except for octaves are slightly out of tune
with regard to the actual harmonic series. For more about equal
temperament, see Tuning Systems. Interestingly, musicians have a
tendency to revert to true harmonics when they can (in other words, when
it is easy to fine-tune each note). For example, an a capella choral group or
a brass ensemble, may find themselves singing or playing perfect fourths
and fifths, "contracted" major thirds and "expanded" minor thirds.

Brass Instruments

The harmonic series is particularly important for brass instruments. A
pianist or xylophone player only gets one note from each key. A string
player who wants a different note from a string holds the string tightly in a
different place. This basically makes a vibrating string of a new length, with
a new fundamental.

But a brass player, without changing the length of the instrument, gets
different notes by actually playing the harmonics of the instrument.
Woodwinds also do this, although not as much. Most woodwinds can get
two different octaves with essentially the same fingering; the lower octave
is the fundamental of the column of air inside the instrument at that
fingering. The upper octave is the first harmonic.

But it is the brass instruments that excel in getting different notes from the
same length of tubing. The sound of a brass instruments starts with
vibrations of the player's lips. By vibrating the lips at different speeds, the
player can cause a harmonic of the air column to sound instead of the
fundamental.

So a bugle player can play any note in the harmonic series of the instrument
that falls within the player's range. Compare these well-known bugle calls
to the harmonic series above.

Bugle Calls

f P| P+ 2 | -
(Ae gd gg tO et
SS ll SC

Although limited by the fact that it can only play

one harmonic series, the bugle can still play many
well-known tunes.

For centuries, all brass instruments were valveless. A brass instrument
could play only the notes of one harmonic series. The upper octaves of the
series, where the notes are close together, could be difficult or impossible to
play, and some of the harmonics sound quite out of tune to ears that expect
equal temperament. The solution to these problems, once brass valves were
perfected, was to add a few valves to the instrument. Three is usually
enough. Each valve opens an extra length of tube, making the instrument a
little longer, and making available a whole new harmonic series. Usually
one valve gives the harmonic series one half step lower than the valveless
intrument, another one whole step lower, and another one and a half steps
lower. The valves can be used at the same time, too, making even more
harmonic series. So a valved brass instrument can find, in the comfortable
middle of its range (its middle register), a valve combination that will give
a reasonably in-tune version for every note of the chromatic scale. (For
more on the history of valved brass, see History of the French Horn. For
more on how and why harmonics are produced in wind instruments, please
see Standing Waves and Wind Instruments)

Note:Trombones use a slide instead of valves to make their instrument
longer. But the basic principle is still the same. At each slide "position", the
instrument gets a new harmonic series. The notes in between the positions
aren't part of the chromatic scale, so they are usually only used for special
effects like glissandos (sliding notes).

Overlapping Harmonic Series in Brass Instruments

1st valve: Harmonic Series one whole step lower bm» 5 rel

——_————=

ba

Mid-range notes available using no valve, 2nd valve alone, or 1st valve alone

————

These harmonic series are for a brass instrument
that has a "C" fundamental when no valves are
being used - for example, a C trumpet.
Remember, there is an entire harmonic series for
every fundamental, and any note can be a
fundamental. You just have to find the brass tube
with the right length. So a trumpet or tuba can get
one harmonic series using no valves, another one
a half step lower using one valve, another one a
whole step lower using another valve, and so on.
By the time all the combinations of valves are
used, there is some way to get an in-tune version
of every note they need.

Exercise:

Problem:

Write the harmonic series for the instrument above when both the first
and second valves are open. (You can use this PDF file if you need
staff paper.) What new notes are added in the instrument's middle
range? Are any notes still missing?

Solution:

Opening both first and second valves gives the harmonic series one-
and-a-half steps lower than "no valves".

A Harmonic Series

New midrange notes:

—==

e

The only midrange note still missing is the ch :
which can be played by adding a third valve, and
holding down the second and third valves at the same time.

Note:The French horn has a reputation for being a "difficult" instrument to
play. This is also because of the harmonic series. Most brass instruments
play in the first few octaves of the harmonic series, where the notes are
farther apart and it takes a pretty big difference in the mouth and lips (the
embouchure, pronounced AHM-buh-sher) to get a different note. The
range of the French horn is higher in the harmonic series, where the notes
are closer together. So very small differences in the mouth and lips can
mean the wrong harmonic comes out.

Playing Harmonics on Strings

String players also use harmonics, although not as much as brass players.
Harmonics on strings have a very different timbre from ordinary string
sounds. They give a quieter, thinner, more bell-like tone, and are usually
used as a kind of ear-catching-special-effect.

Normally when a string player puts a finger on a string, he holds it down
tight. This basically forms a (temporarily) shorter vibrating string, which
then produces an entire harmonic series, with a shorter (higher)

fundamental.

In order to play a harmonic, he touches the string very, very lightly instead.
So the length of the string does not change. Instead, the light touch
interferes with all of the vibrations that don't have a node at that spot. (A
node is a place in the wave where the string does not move back-and-forth.
For example, the ends of the string are both nodes, since they are held in

place.)
String Harmonics

Open strings:

the string vibrates
at all its harmonics
at the same time.

— A Fe FN
Harmonics I \ ' y pad
When a string is touched | “~ x
lightly at a certain spot, i“ |) ‘e

only the harmonics
that have a node
exactly at that spot
can still vibrate.

Nodes

The open string can vibrate
at all these frequencies
at the same time.

A string that is touched lightly
exactly at its midpoint

can only vibrate at the
frequencies that have

anode there. So it will

have a "thinner sound than
the open string. It will also
sound one octave higher
than the open string.

The thinner, quieter sound of "playing harmonics" is caused by the fact that
much of the harmonic series is missing from the sound, which will of
course be heard in the timbre. Lightly touching the string in most spots will
result in no sound at all. It only works at the precise spots that will leave
some of the main harmonics (the longer, louder, lower-numbered ones) free

to vibrate.

Powers, Roots, and Equal Temperament
A review of roots and powers for the music student who wishes to
understand frequency relationships in equal temperament.

You do not need to use powers and roots to discuss music unless you want
to talk about frequency relationships. They are particularly useful when
discussing equal temperament. (See Tuning Systems.)

Powers are simply a shorthand way to write "a certain number times itself
SO many times".

Example:

2*= 2x2x2x2

47= 4x4

Roe St. Siete ote ace Swe

10° = 10 x 10 x 10

Roots are the opposite of powers. They are a quick way to write the idea
"the number that, multiplied by itself so many times, will give this number".

Example:
16 = 2, because 2x2x2x2=16
J125 = 5, because 5x5x5=125

16 = </16 ("the square root of 16") = 4, because 4x4=16

Often one number does not go into another number evenly, so
most roots end up being irrational numbers

(decimals that go on and on with no stopping point)

For example:

¢/3 = 1.7320508...

Roots and powers are relevant to music because equal temperament divides
the octave into twelve equal half steps. A note one octave higher than
another note has a frequency that is two times higher. So if you divide the
octave into twelve equal parts (half steps), the size of each half step is "the
twelfth root of two". (Notice that it is not "2 divided by twelve" or "one
twelfth". For more on this, see Equal Temperament.)

Example:

Of course, you can mix roots and powers in the same math problem.
For example,

( Hie )3 = the fourth root of 16, to the third power
= 2,tothe third power ( because 2 x 2x 2x 2=16)

= §8 (because2x2x2=8)

Following the same steps,

(J9 )° = 81

Do you see why?

Exercise:

Problem:
Using a scientific calculator, find

1. The frequency ratio of a half step (the twelfth root of 2), to the
nearest ten thousandth (four decimal places).

2. The frequency ratio of a perfect fourth (five half steps, or the
twelfth root of 2 raised to the fifth power), to the nearest ten
thousandth.

3. The frequency ratio of a major third (four half steps), to the
nearest ten thousandth.

4. The frequency ratio of an octave.

Solution:

£;1.0595

2. The twelfth root of 2, to the fifth power, is approximately 1.3348

3. The twelfth root of 2, to the fourth power, is approximately
1.2599

4. The twelfth root of 2, to the twelfth power, is 2

Tuning Systems
An overview of music tuning systems

Introduction

The first thing musicians must do before they can play together is "tune".
For musicians in the standard Western music tradition, this means agreeing
on exactly what pitch (what frequency) is an "A", what is a '"B flat" and so
on. Other cultures not only have different note names and different scales,
they may even have different notes - different pitches - based on a different
tuning system. In fact, the modern Western tuning system, which is called
equal temperament, replaced (relatively recently) other tuning systems
that were once popular in Europe. All tuning systems are based on the
physics of sound. But they all are also affected by the history of their music
traditions, as well as by the tuning peculiarities of the instruments used in

To understand all of the discussion below, you must be comfortable with
both the musical concept of interval and the physics concept of frequency.
If you wish to follow the whole thing but are a little hazy on the relationship
between pitch and frequency, the following may be helpful: Pitch;
Acoustics for Music Theory; Harmonic Series I: ‘Timbre and Octaves; and
Octaves and the Major-Minor Tonal System. If you do not know what
intervals are (for example, major thirds and perfect fourths), please see
Interval and Harmonic Series IJ: Harmonics, Intervals and Instruments. If
you need to review the mathematical concepts, please see Musical Intervals,

Meanwhile, here is a reasonably nontechnical summary of the information
below: Modern Western music uses the equal temperament tuning system.
In this system, an octave (say, from C to C) is divided into twelve equally-
spaced notes. "Equally-spaced" to a musician basically means that each of
these notes is one half step from the next, and that all half steps sound like
the same size pitch change. (To a scientist or engineer, "equally-spaced"
means that the ratio of the frequencies of the two notes in any half step is
always the same.) This tuning system is very convenient for some
instruments, such as the piano, and also makes it very easy to change key

without retuning instruments. But a careful hearing of the music, or a look
at the physics of the sound waves involved, reveals that equal-temperament
pitches are not based on the harmonics physically produced by any musical
sound. The "equal" ratios of its half steps are the twelfth root of two, rather
than reflecting the simpler ratios produced by the sounds themselves, and
the important intervals that build harmonies can sound slightly out of tune.
This often leads to some "tweaking" of the tuning in real performances,
away from equal temperament. It also leads many other music traditions to
prefer tunings other than equal temperament, particularly tunings in which
some of the important intervals are based on the pure, simple-ratio intervals
of physics. In order to feature these favored intervals, a tuning tradition may
do one or more of the following: use scales in which the notes are not
equally spaced; avoid any notes or intervals which don't work with a
particular tuning; change the tuning of some notes when the key or mode
changes.

Tuning based on the Harmonic Series

Almost all music traditions recognize the octave. When note Y has a
frequency that is twice the frequency of note Z, then note Y is one octave
higher than note Z. A simple mathematical way to say this is that the ratio
of the frequencies is 2:1. Two notes that are exactly one octave apart sound
good together because their frequencies are related in such a simple way. If
a note had a frequency, for example, that was 2.11 times the frequency of
another note (instead of exactly 2 times), the two notes would not sound so
good together. In fact, most people would find the effect very unpleasant
and would say that the notes are not "in tune" with each other.

To find other notes that sound "in tune" with each other, we look for other
sets of pitches that have a "simple" frequency relationship. These sets of
pitches with closely related frequencies are often written in common
notation as a harmonic series. The harmonic series is not just a useful idea
constructed by music theory; it is often found in "real life", in the real-world
physics of musical sounds. For example, a bugle can play only the notes of
a specific harmonic series. And every musical note you hear is not a single
pure frequency, but is actually a blend of the pitches of a particular
harmonic series. The relative strengths of the harmonics are what gives the

note its timbre. (See Harmonic Series Il: Harmonics, Intervals and

and Wind Instruments for more about how and why musical sounds are
built from harmonic series.)
Harmonic Series on C

e
oe 5 6 7 8 9 10 11 #12 #13 +=«214 15 16

Here are the first sixteen pitches in a harmonic
series that starts on a C natural. The series goes on
indefinitely, with the pitches getting closer and
closer together. A harmonic series can start on any
note, so there are many harmonic series, but every
harmonic series has the same set of intervals and
the same frequency ratios.

What does it mean to say that two pitches have a "simple frequency
relationship"? It doesn't mean that their frequencies are almost the same.
Two notes whose frequencies are almost the same - say, the frequency of
one is 1.005 times the other - sound bad together. Again, anyone who is
accustomed to precise tuning would say they are "out of tune”. Notes with a
close relationship have frequencies that can be written as a ratio of two
small whole numbers; the smaller the numbers, the more closely related the
notes are. Two notes that are exactly the same pitch, for example, have a
frequency ratio of 1:1, and octaves, as we have already seen, are 2:1. Notice
that when two pitches are related in this simple-ratio way, it means that they
can be considered part of the same harmonic series, and in fact the actual
harmonic series of the two notes may also overlap and reinforce each other.
The fact that the two notes are complementing and reinforcing each other in
this way, rather than presenting the human ear with two completely

different harmonic series, may be a major reason why they sound consonant
and "in tune".

Note:Nobody has yet proven a physical basis for why simple-ratio
combinations sound pleasant to us. For a readable introduction to the
subject, I suggest Robert Jourdain's Music, the Brain, and Ecstasy

Notice that the actual frequencies of the notes do not matter. What matters
is how they compare to each other - basically, how many waves of one note
go by for each wave of the other note. Although the actual frequencies of
the notes will change for every harmonic series, the comparative distance
between the notes, their interval, will be the same.

For more examples, look at the harmonic series in [link]. The number
beneath a note tells you the relationship of that note's frequency to the
frequency of the first note in the series - the fundamental. For example, the
frequency of the note numbered 3 in [link] is three times the frequency of
the fundamental, and the frequency of the note numbered fifteen is fifteen
times the frequency of the fundamental. In the example, the fundamental is
a C. That note's frequency times 2 gives you another C; times 2 again (4)
gives another C; times 2 again gives another C (8), and so on. Now look at
the G's in this series. The first one is number 3 in the series. 3 times 2 is 6,
and number 6 in the series is also a G. So is number 12 (6 times 2). Check
for yourself the other notes in the series that are an octave apart. You will
find that the ratio for one octave is always 2:1, just as the ratio for a unison
is always 1:1. Notes with this small-number ratio of 2:1 are so closely
related that we give them the same name, and most tuning systems are
based on this octave relationship.

The next closest relationship is the one based on the 3:2 ratio, the interval of
the perfect fifth (for example, the C and G in the example harmonic series).
The next lowest ratio, 4:3, gives the interval of a perfect fourth. Again,
these pitches are so closely related and sound so good together that their
intervals have been named "perfect". The perfect fifth figures prominently

in many tuning systems. In Western music, all major and minor chords
contain, or at least strongly imply, a perfect fifth. (See ‘Triads and Naming
Triads for more about the intervals in major and minor chords.)

Pythagorean Intonation

The Pythagorean system is so named because it was actually discussed by
Pythagoras, the famous Greek mathematician and philosopher, who in the
sixth century B.C. already recognized the simple arithmetical relationship
involved in intervals of octaves, fifths, and fourths. He and his followers
believed that numbers were the ruling principle of the universe, and that
musical harmonies were a basic expression of the mathematical laws of the
universe. Their model of the universe involved the "celestial spheres"
creating a kind of harmony as they moved in circles dictated by the same
arithmetical relationships as musical harmonies.

In the Pythagorean system, all tuning is based on the interval of the pure
fifth. Pure intervals are the ones found in the harmonic series, with very
simple frequency ratios. So a pure fifth will have a frequency ratio of
exactly 3:2. Using a series of perfect fifths (and assuming perfect octaves,
too, so that you are filling in every octave as you go), you can eventually
fill in an entire chromatic scale.

Pythagorean Intonation

A series of pure perfect fifths...

when the notes are rearranged to
be in the same octave...

gives a scale.

You can continue this series of perfect fifths to get
the rest of the notes of a chromatic scale; the
series would continue F sharp, C sharp, and so on.

The main weakness of the Pythagorean system is that a series of pure
perfect fifths will never take you to a note that is a pure octave above the
note you started on. To see why this is a problem, imagine beginning on a
C. A series of perfect fifths would give: C, G, D, A, E, B, F sharp, C sharp,
G sharp, D sharp, A sharp, E sharp, and B sharp. In equal temperament
(which doesn't use pure fifths), that B sharp would be exactly the same
pitch as the C seven octaves above where you started (so that the series can,
in essence, be turned into a closed loop, the Circle of Fifths). Unfortunately,
the B sharp that you arrive at after a series of pure fifths is a little higher
than that C.

So in order to keep pure octaves, instruments that use Pythagorean tuning
have to use eleven pure fifths and one smaller fifth. The smaller fifth has
traditionally been called a wolf fifth because of its unpleasant sound. Keys
that avoid the wolf fifth sound just fine on instruments that are tuned this
way, but keys in which the wolf fifth is often heard become a problem. To
avoid some of the harshness of the wolf intervals, some harpsichords and
other keyboard instruments were built with split keys for D sharp/E flat and
for G sharp/A flat. The front half of the key would play one note, and the
back half the other (differently tuned) note.

Pythagorean tuning was widely used in medieval and Renaissance times.
Major seconds and thirds are larger in Pythagorean intonation than in equal
temperament, and minor seconds and thirds are smaller. Some people feel
that using such intervals in medieval music is not only more authentic, but
sounds better too, since the music was composed for this tuning system.

More modern Western music, on the other hand, does not sound pleasant
using Pythagorean intonation. Although the fifths sound great, the thirds are
simply too far away from the pure major and minor thirds of the harmonic
series. In medieval music, the third was considered a dissonance and was
used sparingly - and actually, when you're using Pythagorean tuning, it
really is a dissonance - but most modern harmonies are built from thirds
(see Triads). In fact, the common harmonic tradition that includes
everything from Baroque counterpoint to modern rock is often called
triadic harmony.

Some modern Non-Western music traditions, which have a very different
approach to melody and harmony, still base their tuning on the perfect fifth.
Wolf fifths and ugly thirds are not a problem in these traditions, which build
each mode within the framework of the perfect fifth, retuning for different
modes as necessary. To read a little about one such tradition, please see
Indian Classical Music: Tuning and Ragas.

Mean-tone System

The mean-tone system, in order to have pleasant-sounding thirds, takes
rather the opposite approach from the Pythagorean. It uses the pure major
third. In this system, the whole tone (or whole step) is considered to be
exactly half of the pure major third. This is the mean, or average, of the two
tones, that gives the system its name. A semitone (or half step) is exactly
half (another mean) of a whole tone.

These smaller intervals all work out well in mean-tone tuning, but the result
is a fifth that is noticeably smaller than a pure fifth. And a series of pure
thirds will also eventually not line up with pure octaves, so an instrument
tuned this way will also have a problem with wolf intervals.

As mentioned above, Pythagorean tuning made sense in medieval times,
when music was dominated by fifths. Once the concept of harmony in
thirds took hold, thirds became the most important interval; simple perfect
fifths were now heard as "austere" and, well, medieval-sounding. So mean-
tone tuning was very popular in Europe in the 16th through 18th centuries.

But fifths can't be avoided entirely. A basic major or minor chord, for
example, is built of two thirds, but it also has a perfect fifth between its
outer two notes (see Triads). So even while mean-tone tuning was enjoying
great popularity, some composers and musicians were searching for other
solutions.

Just Intonation

In just intonation, the fifth and the third are both based on the pure,
harmonic series interval. Because chords are constructed of thirds and fifths
(see Triads), this tuning makes typical Western harmonies particularly
resonant and pleasing to the ear; so this tuning is often used (sometimes
unconsciously) by musicians who can make small tuning adjustments
quickly. This includes vocalists, most wind instruments, and many string
instruments.

As explained above, using pure fifths and thirds will require some sort of
adjustment somewhere. Just intonation makes two accommodations to
allow its pure intervals. One is to allow inequality in the other intervals.
Look again at the harmonic series.

Major Whole Tone Minor Whole Tone

e 2
Harmonic: 4 5 6 7 8 9 10 11 12

Both the 9:8 ratio and the 10:9 ratio
in the harmonic series are written as
whole notes. 9:8 is considered a
major whole tone and 10:9 a minor
whole tone. The difference between
them is less than a quarter of a
semitone.

As the series goes on, the ratios get smaller and the notes closer together.
Common notation writes all of these "close together" intervals as whole
steps (whole tones) or half steps (semitones), but they are of course all
slightly different from each other. For example, the notes with frequency
ratios of 9:8 and 10:9 and 11:10 are all written as whole steps. To compare
how close (or far) they actually are, turn the ratios into decimals.

Whole Step Ratios Written as Decimals

e 9/8 = 1.125
e 10/9 = 1.111
e 11/10 =1.1

These are fairly small differences, but they can still be heard easily by the
human ear. Just intonation uses both the 9:8 whole tone, which is called a
major whole tone and the 10:9 whole tone, which is called a minor whole
tone, in order to construct both pure thirds and pure fifths.

Note:In case you are curious, the size of the whole tone of the "mean tone"
system is also the mean, or average, of the major and minor whole tones.

The other accommodation with reality that just intonation must make is the
fact that a single just-intonation tuning cannot be used to play in multiple
keys. In constructing a just-intonation tuning, it matters which steps of the
scale are major whole tones and which are minor whole tones, so an
instrument tuned exactly to play with just intonation in the key of C major
will have to retune to play in C sharp major or D major. For instruments
that can tune almost instantly, like voices, violins, and trombones, this is not
a problem; but it is unworkable for pianos, harps, and other other
instruments that cannot make small tuning adjustments quickly.

As of this writing, there was useful information about various tuning
systems at several different websites, including The Development of
Musical Tuning Systems, where one could hear what some intervals sound
like in the different tuning systems, and Kyle Gann's Just Intonation
Explained, which included some audio samples of works played using just
intonation.

Temperament

There are times when tuning is not much of an issue. When a good choir
sings in harmony without instruments, they will tune without even thinking
about it. All chords will tend towards pure fifths and thirds, as well as

seconds, fourths, sixths, and sevenths that reflect the harmonic series.
Instruments that can bend most pitches enough to fine-tune them during a
performance - and this includes most orchestral instruments - also tend to
play the "pure" intervals. This can happen unconsciously, or it can be
deliberate, as when a conductor asks for an interval to be "expanded" or
"contracted".

But for many instruments, such as piano, organ, harp, bells, harpsichord,
xylophone - any instrument that cannot be fine-tuned quickly - tuning is a
big issue. A harpsichord that has been tuned using the Pythagorean system
or just intonation may sound perfectly in tune in one key - C major, for
example - and fairly well in tune in a related key - G major - but badly out
of tune ina "distant" key like D flat major. Adding split keys or extra keys
can help (this was a common solution for a time), but also makes the
instrument more difficult to play. In Western music, the tuning systems that
have been invented and widely used that directly address this problem are
the various temperaments, in which the tuning of notes is "tempered"
slightly from pure intervals. (Non-Western music traditions have their own
tuning systems, which is too big a subject to address here. See Listening to
Balinese Gamelan and Indian Classical Music: Tuning and Ragas for a taste
of what's out there.)

Well Temperaments

As mentioned above, the various tuning systems based on pure intervals
eventually have to include "wolf" intervals that make some keys unpleasant
or even unusable. The various well temperament tunings that were very
popular in the 18th and 19th centuries tried to strike a balance between
staying close to pure intervals and avoiding wolf intervals. A well
temperament might have several pure fifths, for example, and several fifths
that are smaller than a pure fifth, but not so small that they are "wolf" fifths.
In such systems, tuning would be noticeably different in each key, but every
key would still be pleasant-sounding and usable. This made well
temperaments particularly welcome for players of difficult-to-tune
instruments like the harpsichord and piano.

Note: Historically, there has been some confusion as to whether or not well
temperament and equal temperament are the same thing, possibly because
well temperaments were sometimes referred to at the time as "equal
temperament". But these well temperaments made all keys equally useful,
not equal-sounding as modern equal temperament does.

As mentioned above, mean-tone tuning was still very popular in the
eighteenth century. J. S. Bach wrote his famous "Well-Tempered Klavier" in
part as a plea and advertisement to switch to a well temperament system.
Various well temperaments did become very popular in the eighteenth and
nineteenth centuries, and much of the keyboard-instrument music of those
centuries may have been written to take advantage of the tuning
characteristics of particular keys in particular well temperaments. Some
modern musicians advocate performing such pieces using well
temperaments, in order to better understand and appreciate them. It is
interesting to note that the different keys in a well temperament tuning were
sometimes considered to be aligned with specific colors and emotions. In
this way they may have had more in common with various modes and ragas
than do keys in equal temperament.

Equal Temperament

In modern times, well temperaments have been replaced by equal
temperament, so much so in Western music that equal temperament is
considered standard tuning even for voice and for instruments that are more
likely to play using just intonation when they can (see above). In equal
temperament, only octaves are pure intervals. The octave is divided into
twelve equally spaced half steps, and all other intervals are measured in half
steps. This gives, for example, a fifth that is a bit smaller than a pure fifth,
and a major third that is larger than the pure major third. The differences are
smaller than the wolf tones found in other tuning systems, but they are still
there.

Equal temperament is well suited to music that changes key often, is very
chromatic, or is harmonically complex. It is also the obvious choice for
atonal music that steers away from identification with any key or tonality at
all. Equal temperament has a clear scientific/mathematical basis, is very
straightforward, does not require retuning for key changes, and is
unquestioningly accepted by most people. However, because of the lack of
pure intervals, some musicians do not find it satisfying. As mentioned
above, just intonation is sometimes substituted for equal temperament when
practical, and some musicians would also like to reintroduce well
temperaments, at least for performances of music which was composed
with well temperament in mind.

A Comparison of Equal Temperament with the Harmonic
Series

In a way, equal temperament is also a compromise between the Pythagorean
approach and the mean-tone approach. Neither the third nor the fifth is
pure, but neither of them is terribly far off, either. Because equal
temperament divides the octave into twelve equal semi-tones (half steps),
the frequency ratio of each semi-tone is the twelfth root of 2. If you do not
understand why it is the twelfth root of 2 rather than, say, one twelfth,
please see the explanation below. (There is a review of powers and roots in

V2 = a semitone (half step)

2
(2/2 ) = awhole tone (whole step)
(2 2 we a major third (four semitones)
7
W 2)= a perfect fifth (seven semitones)

2 2
(\/2 )=2= an octave (twelve semitones)

In equal temperament,
the ratio of frequencies
in a semitone (half step)
is the twelfth root of

two. Every interval is
then simply a certain
number of semitones.
Only the octave (the
twelfth power of the
twelfth root) is a pure
interval.

In equal temperament, the only pure interval is the octave. (The twelfth
power of the twelfth root of two is simply two.) All other intervals are given
by irrational numbers based on the twelfth root of two, not nice numbers
that can be written as a ratio of two small whole numbers. In spite of this,
equal temperament works fairly well, because most of the intervals it gives
actually fall quite close to the pure intervals. To see that this is so, look at
[link]. Equal temperament and pure intervals are calculated as decimals and
compared to each other. (You can find these decimals for yourself using a
calculator.)

Comparing the Frequency Ratios for Equal Temperament and Pure
Harmonic Series

Equal Temperament Harmonic Series.
Frequency Approximate Frequency
Interval Ratio Difference Ratio
Unison ("%2)° = 1.0000 0.0 1.0000 = 1/1
MinorSecond ('%2)' = 1.0595 0.0314 1.0909 = 12/11
MajorSecond {'72)° = 1.1225 0.0025 1.1250 = 9/8
MinorThird (22)° = 1.1892 0.0108 1.2000 = 6/5
MajorThird ('%2)* = 1.2599 0.0099 1.2500 = 5/4
Perfect Fourth (72)° = 1.3348 0.0015 1.3333 = 4/3
Tritone (72)° = 1.4142 0.0142 1.4000 = 7/5
Perfect Fifth ('%2)’ = 1.4983 0.0017 1.5000 = 3/2
Minor Sixth (1%2)° = 1.5874 0.0126 1.6000 = 8/5
Major Sixth ('72)° = 1.6818 0.0151 1.6667 = 5/3
MinorSeventh ('%2)'° = 1.7818 0.0318 1.7500 = 7/4
Major Seventh (‘72 )'' = 1.8897 0.0564 1.8333 = 11/6
Octave ( 172)" = 2.0000 0.0 2.0000 = 2/1

Look again at [link] to see where pure
interval ratios come from. The ratios for

equal temperament are all multiples of the
twelfth root of two. Both sets of ratios are
converted to decimals (to the nearest ten
thousandth), so you can easily compare
them.

Except for the unison and the octave, none of the ratios for equal
temperament are exactly the same as for the pure interval. Many of them
are reasonably close, though. In particular, perfect fourths and fifths and
major thirds are not too far from the pure intervals. The intervals that are
the furthest from the pure intervals are the major seventh, minor seventh,
and minor second (intervals that are considered dissonant anyway).

Because equal temperament is now so widely accepted as standard tuning,
musicians do not usually even speak of intervals in terms of ratios. Instead,
tuning itself is now defined in terms of equal-temperament, with tunings
and intervals measured in cents. A cent is 1/100 (the hundredth root) of an
equal-temperament semitone. In this system, for example, the major whole
tone discussed above measures 204 cents, the minor whole tone 182 cents,
and a pure fifth is 702 cents.

Why is a cent the hundredth root of a semitone, and why is a semitone the
twelfth root of an octave? If it bothers you that the ratios in equal
temperament are roots, remember the pure octaves and fifths of the
harmonic series.

Frequency Relationships

eo 4 (4x 2) Octaves:
2 (2x2) Higher Frequency =
1 Lower Frequency x 2
*“s ”~ 27/8 Fifths:
9/4 (9/4 x 3/2) Higher Frequency =
3/2 (3/2x 3/2) Lower Frequency x 3/2

Remember that, no matter what note
you start on, the note one octave
higher has 2 times its frequency. Also,
no matter what note you start on, the
note that is a perfect fifth higher has
exactly one and a half times its
frequency. Since each of these
intervals is so many "times" in terms
of frequencies, when you add
intervals, you multiply their
frequencies. For example, a series of
two perfect fifths will give a
frequency that is 3/2 x 3/2 (or 9/4) the
beginning frequency.

Every octave has the same frequency ratio; the higher note will have 2
times the frequency of the lower note. So if you go up another octave from
there (another 2 times), that note must have 2 x 2, or 4 times the frequency
of the lowest note. The next octave takes you up 2 times higher than that, or
8 times the frequency of the first note, and so on.

In just the same way, in every perfect fifth, the higher note will have a
frequency one and a half (3/2) times the lower note. So to find out how

much higher the frequency is after a series of perfect fifths, you would have
to multiply (not add) by one and a half (3/2) every time you went up
another perfect fifth.

All intervals work in this same way. So, in order for twelve semitones (half
steps) to equal one octave, the size of a half step has to be a number that
gives the answer "2" (the size of an octave) when you multiply it twelve
times: in other words, the twelfth root of two. And in order for a hundred
cents to equal one semitone, the size of a cent must be the number that,
when you multiply it 100 times, ends up being the same size as a semitone;
in other words, the hundredth root of the twelfth root of two. This is one
reason why most musicians prefer to talk in terms of cents and intervals
instead of frequencies.

Beats and Wide Tuning

One well-known result of tempered tunings is the aural phenomenon known
as beats. As mentioned above, in a pure interval the sound waves have
frequencies that are related to each other by very simple ratios. Physically
speaking, this means that the two smooth waves line up together so well
that the combined wave - the wave you hear when the two are played at the
Same time - is also a smooth and very steady wave. Tunings that are slightly
off from the pure interval, however, will result in a combined wave that has
an extra bumpiness in it. Because the two waves are each very even, the
bump itself is very even and regular, and can be heard as a "beat" - a very
regular change in the intensity of the sound. The beats are so regular, in
fact, that they can be timed; for equal temperament they are on the order of
a beat per second in the mid range of a piano. A piano tuner works by
listening to and timing these beats, rather than by being able to "hear" equal
temperament intervals precisely.

It should also be noted that some music traditions around the world do not
use the type of precision tunings described above, not because they can't,
but because of an aesthetic preference for wide tuning. In these traditions,
the sound of many people playing precisely the same pitch is considered a
thin, uninteresting sound; the sound of many people playing near the same
pitch is heard as full, lively, and more interesting.

Some music traditions even use an extremely precise version of wide
tuning. The gamelan orchestras of southeast Asia, for example, have an
aesthetic preference for the "lively and full" sounds that come from
instruments playing near, not on, the same pitch. In some types of
gamelans, pairs of instruments are tuned very precisely so that each pair
produces beats, and the rate of the beats is the same throughout the entire
range of that gamelan. Long-standing traditions allow gamelan craftsmen to
reliably produce such impressive feats of tuning.

Further Study
As of this writing:

e Kyle Gann's An Introduction to Historical Tunings is a good source
about both the historical background and more technical information
about various tunings. It also includes some audio examples.

e The Huygens-Fokker Foundation has a very large on-line bibliography
of tuning and temperament.

e Alfredo Capurso, a researcher in Italy, has developed the Circular
Harmonic System (c.ha.s), a tempered tuning system that solves the
wolf fifth problem by adjusting the size of the octave as well as the
fifth. It also provides an algorithm for generating microtonal scales.
You can read about it at the Circular Harmonic System website or
download a paper on the subject. You can also listen to piano
performances using this tuning by searching for "CHAS tuning" at
YouTube.

e A number of YouTube videos provide comparisons that you can listen
to, for example comparisons of just intonation and equal temperament,
or comparisons of various temperaments.

Standing Waves and Musical Instruments
For middle school and up, an explanation of how standing waves in musical
instruments produce sounds with particular pitches and timbres.

What is a Standing Wave?

Musical tones are produced by musical instruments, or by the voice, which,
from a physics perspective, is a very complex wind instrument. So the
physics of music is the physics of the kinds of sounds these instruments can
make. What kinds of sounds are these? They are tones caused by standing
waves produced in or on the instrument. So the properties of these standing
waves, which are always produced in very specific groups, or series, have
far-reaching effects on music theory.

Most sound waves, including the musical sounds that actually reach our
ears, are not standing waves. Normally, when something makes a wave, the
wave travels outward, gradually spreading out and losing strength, like the
waves moving away from a pebble dropped into a pond.

But when the wave encounters something, it can bounce (reflection) or be
bent (refraction). In fact, you can "trap" waves by making them bounce
back and forth between two or more surfaces. Musical instruments take
advantage of this; they produce pitches by trapping sound waves.

Why are trapped waves useful for music? Any bunch of sound waves will
produce some sort of noise. But to be a tone - a sound with a particular
pitch - a group of sound waves has to be very regular, all exactly the same
distance apart. That's why we can talk about the frequency and wavelength
of tones.

Noise

‘on N/I\JD\I

A noise is a jumble of
sound waves. A tone is a
very regular set of waves,
all the same size and same
distance apart.

So how can you produce a tone? Let's say you have a sound wave trap (for
now, don't worry about what it looks like), and you keep sending more
sound waves into it. Picture a lot of pebbles being dropped into a very small
pool. As the waves start reflecting off the edges of the pond, they interfere
with the new waves, making a jumble of waves that partly cancel each other
out and mostly just roils the pond - noise.

But what if you could arrange the waves so that reflecting waves, instead of
cancelling out the new waves, would reinforce them? The high parts of the
reflected waves would meet the high parts of the oncoming waves and
make them even higher. The low parts of the reflected waves would meet
the low parts of the oncoming waves and make them even lower. Instead of
a roiled mess of waves cancelling each other out, you would have a pond of
perfectly ordered waves, with high points and low points appearing
regularly at the same spots again and again. To help you imagine this, here
are animations of a single wave reflecting back and forth and standing
waves.

This sort of orderliness is actually hard to get from water waves, but
relatively easy to get in sound waves, so that several completely different
types of sound wave "containers" have been developed into musical

instruments. The two most common - strings and hollow tubes - will be
discussed below, but first let's finish discussing what makes a good standing
wave container, and how this affects music theory.

In order to get the necessary constant reinforcement, the container has to be
the perfect size (length) for a certain wavelength, so that waves bouncing
back or being produced at each end reinforce each other, instead of
interfering with each other and cancelling each other out. And it really helps
to keep the container very narrow, so that you don't have to worry about
waves bouncing off the sides and complicating things. So you have a bunch
of regularly-spaced waves that are trapped, bouncing back and forth in a
container that fits their wavelength perfectly. If you could watch these
waves, it would not even look as if they are traveling back and forth.
Instead, waves would seem to be appearing and disappearing regularly at
exactly the same spots, so these trapped waves are called standing waves.

Note:Although standing waves are harder to get in water, the phenomenon
does apparently happen very rarely in lakes, resulting in freak disasters.
You can sometimes get the same effect by pushing a tub of water back and
forth, but this is a messy experiment; you'll know you are getting a
standing wave when the water suddenly starts sloshing much higher - right
out of the tub!

For any narrow "container" of a particular length, there are plenty of
possible standing waves that don't fit. But there are also many standing
waves that do fit. The longest wave that fits it is called the fundamental. It
is also called the first harmonic. The next longest wave that fits is the
second harmonic, or the first overtone. The next longest wave is the third
harmonic, or second overtone, and so on.

Standing Wave Harmonics

Fundamental
Ist Harmonic

First Overtone
2nd Harmonic

Second Overtone
3rd Harmonic

Third Overtone
4th Harmonic

And so on...

There is a whole set of standing
waves, Called harmonics, that will
fit into any "container" of a
specific length. This set of waves
is called a harmonic series.

Notice that it doesn't matter what the length of the fundamental is; the
waves in the second harmonic must be half the length of the first harmonic;
that's the only way they'll both "fit". The waves of the third harmonic must
be a third the length of the first harmonic, and so on. This has a direct effect
on the frequency and pitch of harmonics, and so it affects the basics of
music tremendously. To find out more about these subjects, please see
Frequency, Wavelength, and Pitch, Harmonic Series, or Musical Intervals,
Frequency,_and Ratio.

Standing Waves on Strings

You may have noticed an interesting thing in the animation of standing
waves: there are spots where the "water" goes up and down a great deal,
and other spots where the "water level" doesn't seem to move at all. All
standing waves have places, called nodes, where there is no wave motion,
and antinodes, where the wave is largest. It is the placement of the nodes

that determines which wavelengths "fit" into a musical instrument
"container".

Nodes and Antinodes
Nodes

OOOO

\/

Antinodes

As a standing wave
waves back and forth
(from the red to the
blue position), there
are some spots called
nodes that do not
move at all; basically
there is no change, no
waving up-and-down
(or back-and-forth), at
these spots. The spots
at the biggest part of
the wave - where
there is the most
change during each
wave - are called
antinodes.

One "container" that works very well to produce standing waves is a thin,
very taut string that is held tightly in place at both ends. Since the string is
taut, it vibrates quickly, producing sound waves, if you pluck it, or rub it
with a bow. Since it is held tightly at both ends, that means there has to be a
node at each end of the string. Instruments that produce sound using strings

are called chordophones, or simply strings.
Standing Waves on a String

Whole Halves Thirds Fourths and so on...

But not: or:

A string that's held very tightly at both
ends can only vibrate at very particular
wavelengths. The whole string can
vibrate back and forth. It can vibrate in
halves, with a node at the middle of the
string as well as each end, or in thirds,
fourths, and so on. But any wavelength
that doesn't have a node at each end of
the string, can't make a standing wave
on the string. To get any of those other
wavelengths, you need to change the
length of the vibrating string. That is
what happens when the player holds
the string down with a finger, changing
the vibrating length of the string and
changing where the nodes are.

The fundamental wave is the one that gives a string its pitch. But the string
is making all those other possible vibrations, too, all at the same time, so

that the actual vibration of the string is pretty complex. The other vibrations
(the ones that basically divide the string into halves, thirds and so on)
produce a whole series of harmonics. We don't hear the harmonics as
separate notes, but we do hear them. They are what gives the string its rich,
musical, string-like sound - its timbre. (The sound of a single frequency
alone is a much more mechanical, uninteresting, and unmusical sound.) To
find out more about harmonics and how they affect a musical sound, see
Harmonic Series.

Exercise:

Problem:
When the string player puts a finger down tightly on the string,

1. How has the part of the string that vibrates changed?
2. How does this change the sound waves that the string makes?
3. How does this change the sound that is heard?

Solution:

1. The part of the string that can vibrate is shorter. The finger
becomes the new "end" of the string.

2. The new sound wave is shorter, so its frequency is higher.

3. It sounds higher; it has a higher pitch.

When a
finger
holds the
string
down
tightly,

the finger
becomes
the new
end of the
vibrating
part of
the string.
The
vibrating
part of
the string
is shorter,
and the
whole set
of sound
waves it
makes is
shorter.

Standing Waves in Wind Instruments

The string disturbs the air molecules around it as it vibrates, producing
sound waves in the air. But another great container for standing waves
actually holds standing waves of air inside a long, narrow tube. This type of
instrument is called an aerophone, and the most well-known of this type of
instrument are often called wind instruments because, although the
instrument itself does vibrate a little, most of the sound is produced by
standing waves in the column of air inside the instrument.

If it is possible, have a reed player and a brass player demonstrate to you
the sounds that their mouthpieces make without the instrument. This will be
a much "noisier" sound, with lots of extra frequencies in it that don't sound
very musical. But, when you put the mouthpiece on an instrument shaped
like a tube, only some of the sounds the mouthpiece makes are the right

length for the tube. Because of feedback from the instrument, the only
sound waves that the mouthpiece can produce now are the ones that are just
the right length to become standing waves in the instrument, and the
"noise" is refined into a musical tone.

Standing Waves in Wind Instruments

1. Transverse standing waves shown inside tubes
actually represent movement back and forth
between two extremes.

Usually, nodes are shown at closed ends and
2. ae antinodes at open ends. This represents the
air displacement waves; the alr cannot move

back and forth through the closed end,
but it is free to rush back and forth

3. C- through the open tube end.

The three transverse waves above, for example, represent air movement that
goes back and forth between the state on the left and the state on the right
(the shorter the arrow, the less the air in that area is moving) :

—

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Standing Waves in a wind instrument are usually
shown as displacement waves, with nodes at
closed ends where the air cannot move back-and-
forth.

The standing waves in a wind instrument are a little different from a
vibrating string. The wave on a string is a transverse wave, moving the
string back and forth, rather than moving up and down along the string. But
the wave inside a tube, since it is a sound wave already, is a longitudinal
wave; the waves do not go from side to side in the tube. Instead, they form
along the length of the tube.

Longitudinal Waves in Pipes

_ea_. If you consider the standing waves as

1. pressure waves rather than displacement waves,
the nodes would be at the open end!
This is because it is easy to

ae. change the pressure at a closed end,

2. i but impossible to change it at an open end.
Pressure at an open end
will always be room pressure,

ee ae so standing pressure waves alternate between
3. ae ae high and low pressure at closed ends.

The standing waves in the tubes are actually
longitudinal sound waves. Here the displacement
standing waves in [link] are shown instead as
longitudinal air pressure waves. Each wave would
be oscillating back and forth between the state on the
right and the one on the left. See Standing Waves in
Wind Instruments for more explanation.

The harmonics of wind instruments are also a little more complicated, since
there are two basic shapes (cylindrical and conical) that are useful for wind
instruments, and they have different properties. The standing-wave tube of a
wind instrument also may be open at both ends, or it may be closed at one
end (for a mouthpiece, for example), and this also affects the instrument.
Please see Standing Waves in Wind Instruments if you want more
information on that subject. For the purposes of understanding music
theory, however, the important thing about standing waves in winds is this:
the harmonic series they produce is essentially the same as the harmonic
series on a string. In other words, the second harmonic is still half the
length of the fundamental, the third harmonic is one third the length, and so
on. (Actually, for reasons explained in Standing Waves in Wind

Instruments, some harmonics are "missing" in some wind instruments, but
this mainly affects the timbre and some aspects of playing the instrument. It
does not affect the basic relationships in the harmonic series.)

Standing Waves in Other Objects

So far we have looked at two of the four main groups of musical
instruments: chordophones and aerophones. That leaves membranophones
and idiophones. Membranophones are instruments in which the sound is
produced by making a membrane vibrate; drums are the most familiar
example. Most drums do not produce tones; they produce rhythmic "noise"
(bursts of irregular waves). Some drums do have pitch, due to complex-
patterned standing waves on the membrane that are reinforced in the space
inside the drum. This works a little bit like the waves in tubes, above, but
the waves produced on membranes, though very interesting, are too
complex to be discussed here.

Idiophones are instruments in which the body of the instrument itself, or a
part of it, produces the original vibration. Some of these instruments
(cymbals, for example) produce simple noise-like sounds when struck. But
in some, the shape of the instrument - usually a tube, block, circle, or bell
shape - allows the instrument to ring with a standing-wave vibration when
you strike it. The standing waves in these carefully-shaped-and-sized
idiophones - for example, the blocks on a xylophone - produce pitched
tones, but again, the patterns of standing waves in these instruments are a
little too complicated for this discussion. If a percussion instrument does
produce pitched sounds, however, the reason, again, is that it is mainly
producing harmonic-series overtones.

Note: Although percussion specializes in "noise"-type sounds, even
instruments like snare drums follow the basic physics rule of "bigger
instrument makes longer wavelengths and lower sounds". If you can, listen
to a percussion player or section that is using snare drums, cymbals, or
other percussion of the same type but different sizes. Can you hear the

difference that size makes, as opposed to differences in timbre produced by
different types of drums?

Exercise:

Problem:

Some idiophones, like gongs, ring at many different pitches when they
are struck. Like most drums, they don't have a particular pitch, but
make more of a "noise"-type sound. Other idiophones, though, like
xylophones, are designed to ring at more particular frequencies. Can
you think of some other percussion instruments that get particular
pitches? (Some can get enough different pitches to play a tune.)

Solution:
There are many, but here are some of the most familiar:

e Chimes

e All xylophone-type instruments, such as marimba, vibraphone,
and glockenspiel

e Handbells and other tuned bells

e Steel pan drums

Standing Waves and Wind Instruments
The musical sounds of aerophones (woodwinds and brass) are created by
standing waves in the air inside the instruments.

Introduction

A wind instrument makes a tone when a standing wave of air is created
inside it. In most wind instruments, a vibration that the player makes at the
mouthpiece is picked up and amplified and given a pleasant timbre by the
air inside the tube-shaped body of the instrument. The shape and length of
the inside of the tube give the sound wave its pitch as well as its timbre.

You will find below a discussion of what makes standing waves in a tube,
wind instruments and the harmonic series, and the types of tubes that can be
used in musical instruments. This is a simplified discussion to give you a
basic idea of what's going on inside a wind instrument. Mathematical
equations are avoided, and all the complications - for example, what
happens to the wave when there are closed finger holes in the side of the
tube - are ignored. Actually, the physics of what happens inside real wind
instruments is so complex that physicists are still studying it, and still don't
have all the answers. If you want a more in-depth or more technical
discussion, there are some recommendations below.

If you can't follow the discussion below, try reviewing Acoustics for Music

Wind Instruments: Some Basics

What Makes the Standing Waves in a Tube

As discussed in Standing Waves and Musical Instruments, instruments
produce musical tones by trapping waves of specific lengths in the
instrument. It's pretty easy to see why the standing waves on a string can
only have certain lengths; since the ends of the strings are held in place,
there has to be a node in the wave at each end. But what is it that makes
only certain standing waves possible in a tube of air?

To understand that, you'll have to understand a little bit about what makes
waves in a tube different from waves on a string. Waves on a string are
transverse waves. The string is stretched out in one direction (call it "up and
down"), but when it's vibrating, the motion of the string is in a different
direction (call it "back and forth"). Take a look at this animation. At the
nodes (each end, for example), there is no back and forth motion, but in
between the nodes, the string is moving back and forth very rapidly. The
term for this back-and-forth motion is displacement. There is no
displacement at a node; the most displacement happens at an antinode.
Transverse Motion on a String

Antinode:
most displacement

Node: Other points:
no displacement some displacement,
but less than antinode

The standing waves of air in a tube are not transverse waves. Like all sound
waves, they are longitudinal. So if the air in the tube is moving in a certain
direction (call it "left and right"), the vibrations in the air are going in that
same direction (in this case, they are rushing "left and right").

But they are like the waves on a string in some important ways. Since they
are standing waves, there are still nodes - in this case, places where the air
is not rushing back and forth. And, just as on the string, in between the
nodes there are antinodes, where the displacement is largest (the air is
moving back and forth the most). And when one antinode is going in one
direction (left), the antinodes nearest it will be going in the other direction
(right). So, even though what is happening is very different, the end result
of standing waves "trapped" in a tube will be very much like the end result
of standing waves "trapped" on a string: a harmonic series based on the tube
length.

There will be more on that harmonic series in the next section. First, let's
talk about why only some standing waves will "fit" in a tube of a particular
length. If the tube were closed on both ends, it's easy to see that this would
be a lot like the wave on the string. The air would not be able to rush back

and forth at the ends, so any wave trapped inside this tube would have to
have nodes at each end.

Note:It's very difficult to draw air that is rushing back and forth in some
places and standing still in other places, so most of the figures below use a
common illustration method, showing the longitudinal waves as if they are
simultaneously the two maximum positions of a transverse wave. Here is
an animation that may give you some idea of what is happening in a
longitudinal standing wave. As of this writing, there was a nice Standing
Waves applet demonstration of waves in tubes. Also, see below for more
explanation of what the transverse waves inside the tubes really represent.

Fully Closed Tube

The standing
waves inside the
tube represent
back-and-forth
motion of the air.
Since the air can't
move through the
end of the tube, a
closed tube must
have a node at
each end, just like
a string held at
both ends.

Now, a closed tube wouldn't make a very good musical instrument; it
wouldn't be very loud. Most of the sound you hear from an instrument is

not the standing wave inside the tube; the sound is made at the open ends
where the standing waves manage to create other waves that can move
away from the instrument. Physicists sometimes study the acoustics of a
tube closed at both ends (called a Kundt tube), but most wind instruments
have at least one open end. An instrument that is open at both ends may be
called open-open, or just an open tube instrument. An instrument that is
only open at one end may be called open-closed, or a closed tube or
stopped tube instrument (or sometimes semi-closed or half-closed). This
is a little confusing, since such instruments (trumpets, for example) still
obviously have one open end.

Now, there's nothing stopping the air from rushing back and forth at the
open end of the tube. In fact, the waves that "fit" the tube are the ones that
have antinodes at the open end, so the air is in fact rushing back and forth
there, causing waves (at the same frequency as the standing wave) that are
not trapped in the instrument but can go out into the room.

Open-Open and Open-Closed Tubes

gee aes

There must be a

(displacement)

antinode at any
open end of a tube.

What is it that requires the waves to have an antinode at an open end? Look
again at the animation of what is happening to the air particles in the
standing wave. The air at the nodes is not moving back and forth, but it is
piling up and spreading out again. So the air pressure is changing a lot at
the nodes. But at the antinodes, the air is moving a lot, but it is moving back
and forth, not piling up and spreading out. In fact, you can imagine that
same wave to be an air pressure wave instead of an air displacement wave.
It really is both at the same time, but the pressure wave nodes are at the

same place as the displacement antinodes, and the pressure antinodes are at
the same place as the displacement nodes.
An Air Displacement Wave is also an Air Pressure Wave

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The nodes of the displacement wave, where
the air is not rushing back-and-forth but is
doing the most piling-up-and-spreading-out,
are the antinodes of the pressure wave. The
antinodes of the displacement wave, where
the air is rushing back-and-forth the most,
but is not piling up or spreading out at all,
are the nodes of the pressure wave. Both
waves must have exactly the same
frequency, of course; they are actually just
two aspects of the same sound wave.

At an open end of the tube, there is nothing to stop the air rushing in and
out, and so it does. What the air cannot do at the open end is build up any
pressure; there is nothing for the air to build up against, and any drop in
pressure will just bring air rushing in from outside the tube. So the air
pressure at an open end must remain the same as the air pressure of the
room. In other words, that end must have a pressure node (where the air
pressure doesn't change) and (therefore) a displacement antinode.

Note:Since being exposed to the air pressure outside the instrument is what
is important, the "open end" of a wind instrument, as far as the sound
waves are concerned, is the first place that they can escape - the first open
hole. This is how woodwinds change the length of the wave, and the pitch
of the note. For more on this, please see Wind Instruments — Some Basics.

Harmonic Series in Tubes

As explained in the previous section, the standing waves in a tube must
have a (displacement) node at a closed end and an antinode at an open end.
In an open-open tube, this leads to a harmonic series very similar to a
harmonic series produced on a string that's held at both ends. The
fundamental, the lowest note possible in the tube, is the note with a
wavelength twice the length of the tube (or string). The next possible note
has twice the frequency (half the wavelength) of the fundamental, the next
three times the frequency, the next four times, and so on.

Allowed Waves in an Open Tube

Fundamental a re Wavelength = 2 x tube length
se ee Frequency = 2 x Fundamental
ee Frequency = 3 x Fundamental
i i a a Frequency = 4x Fundamental

These are the first four harmonics allowed in an
open tube. Any standing wave with a
displacement antinode at both ends is allowed,
but the lower harmonics are usually the easiest
to play and the strongest harmonics in the
timbre.

But things are a little different for the tube that is closed at one end and
open at the other. The lowest note that you might be able to get on sucha
tube (a fundamental that is unplayable on many instruments) has a
wavelength four times the length of the tube. (You may notice that this
means that a stopped tube will get a note half the frequency - an octave
lower - than an open tube of the same length.) The next note that is possible
on the half-closed tube has three times the frequency of the fundamental,
the next five times,and so on. In other words, a stopped tube can only play
the odd-numbered harmonics.

Allowed Wavelengths in a Stopped Tube

Fundamental ———— Wavelength = 4x tube length

a,

ae <ai> ae Frequency = 7 x Fundamental

Again, these are the lowest (lowest pitch and
lowest frequency) four harmonics allowed. Any
wave with a displacement node at the closed end

and antinode at the open end is allowed. Note

that this means only the odd-numbered
harmonics "fit".

Note:All of the transverse waves in [link], [link], [link], and [link |
represent longitudinal displacement waves, as shown in [link]. All of the
harmonics would be happening in the tube at the same time, and, for each
harmonic, the displacement ({link]) and pressure waves ({link]) are just
two different ways of representing the same wave.

Displacement Waves

1. =— Transverse standing waves shown inside tubes
actually represent movement back and forth

between two extremes.

Usually, nodes are shown at closed ends and
2. Se antinodes at open ends. This represents the
air displacement waves; the alr cannot move

back and forth through the closed end,
but it is free to rush back and forth

3. co. through the open tube end.

The three transverse waves above, for example, represent air movement that
goes back and forth between the state on the left and the state on the right
(the shorter the arrow, the less the air in that area is moving) :

Oo ides Roe dow atl ob —e, [66-070 —
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Here are the first three possible harmonics in a
closed-open tube shown as longitudinal
displacement waves.

Pressure Waves

If you consider the standing waves as

pressure waves rather than displacement waves,

the nodes would be at the open end!

This is because it is easy to

a. change the pressure at a closed end,

2. ee but impossible to change it at an open end.
Pressure at an open end
will always be room pressure,

ea oe so standing pressure waves alternate between
3. Se eS: high and low pressure at closed ends.

1.

Here are those same three waves shown as pressure
waves.

Basic Wind Instrument Tube Types

The previous section shows why only the odd-numbered harmonics "fit" in
a cylinder-shaped tube, but that is not the whole story. There is one other
tube shape that works well for wind instruments, and it abides by slightly
different rules.

Just as on a string, the actual wave inside the instrument is a complex wave
that includes all of those possible harmonics. A cylinder makes a good
musical instrument because all the waves in the tube happen to have simple,
harmonic-series-type relationships. This becomes very useful when the
player overblows in order to get more notes. As mentioned above,
woodwind players get different notes out of their instruments by opening
and closing finger holes, making the standing wave tube longer or shorter.
Once the player has used all the holes, higher notes are played by
overblowing, which causes the next higher harmonic of the tube to sound.
In other words, the fundamental of the tube is not heard when the player
"overblows"; the note heard is the pitch of the next available harmonic
(either harmonic two or three). Brass players can get many different
harmonics from their instruments, and so do not need as many fingerings.
(Please see Harmonic Series and Wind Instruments — Some Basics for more
on this.)

For most possible tube shapes, a new set of holes would be needed to get
notes that are in tune with the lower set of notes. But a couple of shapes,
including the cylinder, give higher notes that are basically in tune with the
lower notes using the same finger holes (or valves). (Even so, some extra
finger holes or an extra slide or valve is sometimes necessary for good
tuning.) One other possible shape is basically not used because it would be
difficult to build precisely and unwieldy to play. (Basically, it has to flare
rapidly, at a very specific rate of flare. The resulting instrument would be

unwieldy and impractical. Please see John S. Rigden's Physics and the
Sound of Music, as cited below for more on this.)

Cylindrical

Conical

The two shapes that are
useful for real wind
instruments are the

cylinder and the cone.
Most real wind
instruments are a
combination of
cylindrical and conical
sections, but most act as
(and can be classified
as) either cylindrical
bore or conical bore
instruments.

The other tube shape that is often used in wind instruments is the cone. In
fact, most real wind instruments are tubes that are some sort of combination
of cylindrical and conical tubes. But most can be classified as either
cylindrical or conical instruments.

The really surprising thing is that stopped-tube instruments that are
basically conical act as if they are open-tube cylindrical instruments.

Note:The math showing why this happens has been done, but I will not go
into it here. Please see the further reading, below for books with a more
rigorous and in-depth discussion of the subject.

Compare, for example, the clarinet and the saxophone, woodwinds with
very similar mouthpieces. Both instruments, like any basic woodwind, have
enough finger holes and keys to play all the notes within an octave. To get
more notes, a woodwind player overblows, blowing hard enough to sound
the next harmonic of the instrument. For the saxophone, a very conical
instrument, the next harmonic is the next octave (two times the frequency of
the fundamental), and the saxophonist can continue up this next octave by
essentially repeating the fingerings for the first octave. Only a few extra
keys are needed to help with tuning.

The clarinet player doesn't have it so easy. Because the clarinet is a very
cylindrical instrument, the next harmonic available is three times the
frequency, or an octave and a fifth higher, than the fundamental. Extra holes
and keys have to be added to the instrument to get the notes in that missing
fifth, and then even more keys are added to help the clarinetist get around
the awkward fingerings that can ensue. Many notes have several possible
fingerings, and the player must choose fingerings based on tuning and ease
of motion as they change notes.

So why bother with cylindrical instruments? Remember that an actual note
from any instrument is a very complex sound wave that includes lots of
harmonics. The pitch that we hear when a wind instrument plays a note is
(usually) the lowest harmonic that is being produced in the tube at the time.
The higher harmonics produce the timbre, or sound color, of the instrument.
A saxophone-shaped instrument simply can't get that odd-harmonics
clarinet sound.

The shapes and sounds of the instruments that are popular today are the
result of centuries of trial-and-error experimentation by instrument-makers.
Some of them understood something of the physics involved, but the actual
physics of real instruments - once you add sound holes, valves, keys,
mouthpieces, and bells - are incredibly complex, and theoretical physicists
are still studying the subject and making new discoveries.

Further Reading

e Alexander Wood's The Physics of Music (1944, The Sherwood Press)
is a classic which includes both the basics of waves in a pipe and
information about specific instruments.

e John Backus' The Acoustical Foundations of Music (1969, W.W.
Norton and Company) also goes into more detail on the physics of
specific instruments.

e John S. Rigden's Physics and the Sound of Music (1977, John Wiley
and Sons) includes most of the math necessary for a really rigorous,
complete explanation of basic acoustics, but is (in my opinion) still
very readable.

e Arthur H. Benade's Fundamentals of Musical Acoustics is a more
technical textbook that gives some idea of how acoustical experiments
on instruments are designed and carried out. Those who are less
comfortable with the science/engineering aspect of the subject may
prefer the two very thorough articles by Benade in:

e The Physics of Music (W. H. Freeman and Co.), a collection of
readings from the periodical Scientific American.